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The Right to the Beach? Urban Renewal, Public Space Policing and the Definition of a Beach Public in Postwar Los Angeles, 1940s-1960s Elsa DEviEnnE

Cet article montre d'abord comment, après la Seconde Guerre mondiale, une coalition informelle rassemblant des promoteurs immobiliers et l'élite politique locale utilise des stratégies de l'ordre de la planification urbaine et du contrôle policier afin de faire des plages de la ville de Los Angeles un terrain de jeux réservé à la classe moyenne blanche. La première partie de l'article décrit les efforts des urbanistes, des ingénieurs et de l'élite politique et économique afin de mener à bien une campagne de modernisation du littoral qui entraîne notamment la destruction de lieux de loisirs jugés délabrés. Vastes, propres, et dotées d'équipements neufs, ces plages « modernes » doivent attirer touristes, résidents aisés et investisseurs. Ensuite, l'article met en évidence la manière dont la modernisation urbaine sert de justification pour mener des campagnes de harcèlement policier à l'égard des « indésirables », en particulier les homosexuels, nombreux alors à fréquenter des établissements de bains et bars de la côte, et les athlètes de « Muscle Beach », lieu d'entraînement et de performances sportives. La notion de « droit à la ville », telle que développée par Don Mitchell, est utilisée afin de montrer que cette tentative de « sauver » les plages du déclin a pour principale conséquence de restreindre le « droit à la plage » aux seuls baigneurs blancs de la classe moyenne.

t mid-twentieth century, the beaches of Los Angeles-long a key feature of the California dream, born along with the state itself-were in poor condition. In its March 1950 bulletin, the Shoreline Planning Association, an organization dedicated to the development and protection of the California coast, reported that the region's public beaches "call [ed] up in the minds of many people an unpleasant picture […]: over-crowded, littered up, inadequate parking, very noisy, small and unsanitary facilities, the whole nourished by smelly ramshackle hot dog stands skilled in the preparation of dubious hamburgers." [START_REF]Los Angeles Plans Model Facility[END_REF] Just a year earlier, the city planning firm Madigan and Hyland had reached a similar conclusion in a report commissioned by the county of Los Angeles: "The local beaches offer the prospective visitor traffic jams, parking problems, crowded conditions, inadequate and low-grade food stands, honkey-tonk amusements and sand in the shoes." [START_REF]Recreational Development of the Los Angeles Area Shoreline: an Engineering and Economic Report to the Mayor and the City Council[END_REF] With 30 % of the shoreline within Los Angeles County publically owned, the region offered much space to relax for the potential beachgoer, [START_REF] Warren | California State's Beach Program[END_REF] especially compared to Eastern states where the proportion of seashore open to the general public could be as low as 3 %. [START_REF]This figure concerns the Massachusetts shoreline in 1950. See "State Beaches for Massachusetts[END_REF] Yet, according to these reports, the city's public beaches were in such a degraded condition that they might as well have been private.

In the years following the end of the Second World War, the Los Angeles coastline became the focus of many such articles, reports, and plans lamenting the sad state of the beaches. In descriptions eerily similar to contemporaneous assessments of the crowded slums that characterized the central districts of many northern and Midwestern cities, they painted a disheartening picture of beaches needing to be "cleared" from "tenements" and "blight." But these documents also planned for and announced the future. Indeed, in the 1950s and 1960s urban planners, local business and political elites, along with engineers worked together to transform the city's public shoreline into a modern playground, cleaning up miles of dirty beaches, and destroying old piers and barracks. In other words, just like the innercity slums of many industrial American cities, the beaches of Los Angeles experienced a rapid and intense phase of urban renewal.

While the restructuration of postwar city centers inspired urban planners to destroy the old to make space for the new, Disneyland, a new kind of family-friendly amusement park that opened in Orange County in 1955, provided the major source of inspiration for deciding what exactly should be built on the recently bulldozed beaches. By building an amusement park that was clean, orderly, homogeneous, and devoid of any sensual or dubious recreational activities, Walt Disney redefined mainstream leisure for the suburban era. In the postwar years, advocates of beach modernization used both the principles that Disney had developed in his theme park and the precepts of urban renewal to invent a new model of public seashore amusement, one that was intended to appeal to the so-called "respectable public." By ridding beaches of the relics of traditional working-class leisure spaces-old amusement parks, piers, and dance-halls-and by building improved beaches with family-friendly attractions and plenty of room for parking, beach modernizers hoped to keep the white middle-and upper-class families on the shore.

But in order to attract this specific social group-the "respectable" public-beaches also had to be more tightly controlled and policed. While the region's beaches had historically been places of relative freedom (at least for white Angelenos) compared to other public spaces, they became more strictly regulated after the Second World War. In the context of the expansion of the suburbs and the domestic revival of the 1950s, public urban space stirred up strong feelings of fear and anxiety. 5 In order to maintain the "proper" public on the beaches, any beachgoer labeled as "undesirable" had to go. On the beach, where semi-naked bodies are in close proximity to one another, the police defined the "undesirables" based on gender and sexual norms. Starting in the mid-1950s, gay men and lesbians, as well as male and female athletes, whose bodies or choices of activity challenged traditional gender expectations, were the object of surveillance and harassment by the police.

This paper will analyze how, in the postwar period, the combination of urban renewal techniques and increased policing dramatically transformed the Los Angeles coastline and limited the "right to the beach" to the white middle-class family. Using Don Mitchell's discussion of French Marxist philosopher Henri Lefebvre's idea of "the right to the city," I will look at the ways in which the local urban elite produced, through design and surveillance, a seemingly public beach, which only served the needs of a specific group. In the past fifteen years, many scholars have investigated the politics of exclusion that have shaped and reshaped leisure spaces in contemporary American cities. In particular, the history of beach and 5. Historian Elaine Tyler May has described how the Cold War on the Homefront generated a "domestic revival," which contributed to lowering the age of marriage and increasing birthrates, and promoted the home as a "bastion of safety in an insecure world." In contrast, public space became associated with threats coming from a variety of enemies, including anybody who did not conform to social and sexual norms. (May 9). swimming-pool segregation has generated important scholarship, which has underlined the centrality of leisure in the history of the civil rights movement (Wiltse; Wolcott, 2012; Kahrl). More recently, studies focusing on Eastern coastal regions have shown how wealthy shoreline homeowners have managed throughout the twentieth century to prevent federal and state government officials from opening "their" private beaches to the general public and how their tactics, which involved building fortifications on the shores, exacerbated the ecological fragility of the coastline. (Schlichting; Kahrl) My paper contributes to this discussion but departs from existing studies by focusing on the sometimes more covert role played by planning and policing in the exclusion of certain groups from urban recreational spaces. Indeed, in contrast to the violence that erupted when black beachgoers attempted to desegregate beaches, amusement parks, and swimming pools in Chicago, Buffalo, or Biloxi, the ways in which the Los Angeles beach public was filtered remained essentially invisible (Patterson Smith; Wolcott, 2006; Fisher) More specifically, I will show how the combination of beach planning and policing in postwar Los Angeles contributed to the construction of the beach as an exclusive space for the white middle-class nuclear family. This process was crucial in determining the fate of the city's beaches: without extensive planning and policing, many famous urban beaches opened to the public (such as Coney Island or Atlantic City), stopped attracting the middle and upper classes, and were commonly viewed as dwindling attractions in the 1960s and 1970s. 6 Yet by "saving" the public beaches of Los Angeles from urban decline, the men and women behind these initiatives also managed to exclude many Angelenos from the shores.

Urban Renewal on the Beach

Urban renewal corresponds to a two-decade period in postwar urban planning, when planners and political elites took on the task of remaking the industrial cities of America into rational and car-friendly metropolises. Starting in 1949 with the Housing Act, which authorized federally subsidized slum clearance and urban redevelopment, and ending in the early 1970s, when critics of big government and large-scale planning multiplied, this period saw the destruction of entire neighborhoods and their rapid replacement by gleaming office towers, highways, stadiums, and high-rise apartment buildings.

Recent studies have increased and complicated our understanding of this period (Zipp; Klemek; Avila & Rose; Wakeman), but historians have so far largely neglected the impact of these processes outside of inner-city areas. By focusing on efforts to modernize Los Angeles beaches between the late 1940s and the 1960s, my intention is to expand the traditional vision of postwar urban renewal and its impact on everyday life in American cities. In many ways, urban renewal on the beach followed the same precepts and sequence as its equivalent in the city. First, the vocabulary used to describe the shoreline was comparable to the discourses adopted by urban modernization advocates: as with the slums of the inner-city, beaches in Los Angeles were accused of being dirty, blighted, congested, and chaotic. As early as 1941, a reporter for the LA Daily News in charge of assessing the state of the city's coastline compared the beaches to "seaside eyesores." 7 Experts agreed: according to the Chief of the State Division of beaches and parks in 1948, the state's shoreline was "a sorry mess" 8 that needed great improvements. Moreover, like in the city proper, the problems facing urban beaches included lack of hygiene, the presence of dilapidated buildings on the sand, over-crowdedness, and barriers to accessibility. In a final echo of the situation of inner-city slums, the city's solution was to destroy the old to make room for the new: on "glamorous Santa Monica beach," planners hoped to clear out the "tenement style facilities" that stood alongside the lavish homes of Hollywood movie stars in order to build "shelters, promenades, play apparatus, picnic tables and stoves." 9 The different phases of beach modernization also mirrored what happened in the rest of the city. Whereas beach associations had been formed as early as the 1930s in order to improve the region's shoreline, real efforts and funding came only in the late 1940s. The Second World War initiated a particularly prolific period of shoreline planning with California emerging as the leading state in that domain. Between 1944 and 1946, six different plans were produced, drawing up the future of the California coastline, among which five proposals were specifically dedicated to Los Angeles. In 1946, a master plan for the beaches of the metropolitan region was approved and, a few years later, the county hired the urban planning firm Madigan and Hyland to write a report on the plan's feasibility. 10 In the 1950s and 1960s, most of 7. Daily News LA, July 28, 1941, 3. 8. Los Angeles Times, September 25, 1948, 14. 9. Daily News LA, July 30, 1941, 3. As the caption explained, this particular section of Santa Monica Beach, where teenagers liked to gather, was representative of the many deficiencies of the region's beaches (lack of modern facilities, crowdedness, etc.). Picturing semi-naked teenagers in close proximity, at a moment when juvenile delinquency was a hot topic among parents and politicians, was also strategic: by using this photograph, the association alluded to the fact that dirty and crowded beaches contributed to pre-marital sex and delinquent behaviors in general.

the proposals contained in the report were implemented. Yet, by the end of the 1960s beach modernization efforts-just like their equivalents in the inner-city neighborhoods-had died down, falling prey to critics denouncing the construction of rows of Miami-type high-rise buildings on the coast and the displacement of thousands of residents, most of them too poor to relocate anywhere near the beach.

A final similarity between the shore and the city is that the coalition behind beach modernization resembled the group behind urban redevelopment: city officials, planners, politicians, and business leaders were all pushing for beach improvements11 . In the city and on the beach, the goal was to lure wealthy city-dwellers and investments to Los Angeles during an age of metropolitan competition for capital. By developing modern beaches along the coast, beach modernization advocates hoped to establish Los Angeles' status as a world-class city where both work and leisure could be pursued. Moreover, in the context of the expansion of the suburbs and the decline of the downtown neighborhoods, the objective was to make sure that the beach communities would remain attractive to upper-and middle-class white suburbanites.

While the vocabulary, the chronology, and the goals of beach modernization were quite similar to that of urban redevelopment, the actual unfolding of these visions differed. The beaches, being partially of "natural" origin, had to undergo specific changes in order to be suitable sites for modernization and construction. First, for beaches to be attractive in the postwar era, they had to be clean. In Los Angeles, this meant that the issue of water and sand pollution had to be solved. Throughout the 1930s and 1940s, the decrepit sewage treatment plant located at Hyperion did not properly treat the city's wastewater. Raw sewage was dumped directly into the sea at Hyperion beach before contaminating the entire bay. In 1942, a state report stated that intestinal diseases were known to be three times as frequent in proportion to the population in the adjoining shore area as they were elsewhere in the state. 12 The following year, quarantine was established on 10 miles of the city's coastline, including the popular Venice beaches. While Angelenos had bathed for decades in questionably clean water, this situation had to be fixed if officials wanted middle-class visitors to continue frequenting the shore. In the decade following the war, Americans had become increasingly concerned with hygiene and cleanliness. As women left their war-industry jobs-often under societal pressure-they found themselves in large suburban houses with modern electrical appliances that were supposed to help them in their daily chores. Yet this is precisely in that period that new standards of cleanliness were elaborated both as a means to justify the suburban gender binary, which relied on women's presence at home, and to support consumer spending. Indeed, according to historian Suellen Hoy, the American "culture of cleanliness" reached its peak in the 1950s (Hoy 151). As a response to these needs, the beach modernization program included the construction of a brand new sewage disposal plant at Hyperion Beach in 1950, allowing the quarantine to be lifted. 13 But the beaches could not remain clean if they were crowded. Moreover, crowded beaches, like the ones photographer Robert Weegee had captured in his famous Coney Island photographs, were associated with a workingclass form of leisure that had lost its appeal after the war. The 1950s white middle-class family expected a semblance of intimacy, even on a public beach. The modernization supporters responded to that problem by devising and carrying out a vast artificial sand nourishment project on the coast. As early as the 1930s, engineers had developed sand nourishment techniques to rebuild eroded beaches. Yet when Los Angeles engineers envisioned a huge ocean fill of 56,000,000 cubic yards of sand extracted from nearby dunes, many doubted this could be done. After a first successful experiment conducted during the war, when trucked-sand was deposited on the beach, several sand nourishment operations followed in the 1950s, resulting in the vast enlargement of the city's beaches. By the early 1960s, some beaches had been widened up to three times their original sizes. 14 The third step of the modernization program saw the destruction of the deteriorating amusement piers, sideshows, restaurants, nickelodeons, and dance-halls that had been built at the beginning of the century to attract tourists to the coast. Seashore amusement piers had once symbolized modernity and liberation from the strictures of Victorian culture: in the 1950s they became associated with grime, dirt, deviant sexuality, and implicitly with working-class youth and minority Angelenos (Kasson; Peiss). Films of that period reinforced and reflected that connection: Los Angeles piers were painted as the favorite hangouts of criminals, con artists, and racially ambiguous seductresses. One of the first scenes of the movie Mildred Pierce (M. Curtiz, 1945) was shot on a pier where shady bars and sideshows lit up the dark night. This is where Mildred comes to contemplate death. Later in These two pictures entitled "Before and After" were published in California Coast, the official newsletter of the Shoreline Planning Association. The juxtaposition of these two images-one showing ocean water, the other a broad sandy beach-and the description indicating that both were taken at the same place but nine months apart were meant to showcase the striking enlargement of the beaches that was taking place all over the region. In the postwar period, engineers specialized in coastal constructions developed innovative techniques to create or enlarge beaches. These two photographs were taken at Playa Del Rey, a popular Los Angeles beach that was dramatically enlarged in the late 1940s and early 1950s.

the movie, this is also where her daughter degrades herself by performing in a dive bar in front of an inebriated male audience. In T-Men (A. Mann, 1947), another Los Angeles-based film noir of that period, the investigation of a counterfeiting ring leads the main character on the Ocean Park pier into a gambling joint where a Mexican waitress attempts to seduce him.

In Quicksand (I. Pichel, 1950), femme fatale Vera takes Dan, a young automobile mechanic, to Joyland, a Santa Monica Pier penny arcade, where they meet the owner, a seedy character. Later in the movie, Dan, with Vera's encouragements, attempts to break into Joyland and steal the owner's car. In the popular culture of the 1940s and 1950s, the piers and beach sideshows were the favorite haunt of "undesirable" characters. For the beach modernization coalition, those had to go. By destroying amusements that had long been the favorite part of the beach experience for working-class visitors and the youth, the "beach lobby" was directly affecting the kind of public that patronized the coastline. In 1950, the Shoreline Planning Association proudly proclaimed: "Gone are the flimsy wooden bathhouses, the grimy hot-dog stands which saturated the air with the characteristic odor of very stale grease and the inadequate sanitary facilities." 15 The systematic mention of the dirty food stands and their associated stench evinces the classist dimension of the modernization efforts: the goal was to get rid of the working-class elements of the beach in order to appeal to middle-class tastes.

These newly-widened, empty beaches allowed the planners to go through with the next step of their plan: adapting the coastline to the postwar suburbanite values of cleanliness, privacy, and respectability. Indeed, in the 1950s and 1960s, a growing share of American families benefited from the booming economy and gained access to a middle-class suburban lifestyle that used to be reserved to a privileged few. These socioeconomic transformations not only had consequences on the nation's urban landscape-with the onset of the so-called "white flight from the inner-cities to the suburbs-but also on cultural values, with a renewed emphasis on family entertainment, traditional gender roles, and homeownership. Beach modernizes thus hoped that building modern accommodations and familyfriendly recreational spaces on the sand would prevent the white middle class from "fleeing" the beaches as they had "fled" the inner-city districts. In doing so, planners were taking their cues from the new model of leisure that had emerged in 1955 in Orange County: Disneyland. 16 The famous amusement park was imagined as the exact opposite of the beach amusement parks built in reference to Coney Island at the beginning of the century. Walt Disney had specifically refused to locate his park at the beach, fearing that people might show up in their bathing suits. Moreover, the general atmosphere of the beach parks did not fit in with his vision. In contrast to the mixed crowds that lingered around sexualized sideshows and gambled in the dark corners of the ocean piers, Disneyland presented "a controlled landscape that orchestrated the movement and vision of park visitors" (Avila 106). The park was dedicated to the middle-class family and every amusement had to reflect the family-friendly values upheld by Walt Disney. There was no place for dubious beach attractions that promised to "reveal the secrets of sex." 17 Moreover, to make sure that the park would remain inaccessible to carless people (teenagers in particular) Disneyland was built far away from the city and public transportation. Finally, in contrast to the amusement piers, which contained disparate shows with differing prices and styles, Disneyland was self-contained: a single price was paid for the entry into the park, where everything, from the color scheme to the friendly and well-kempt employees, was standardized.

The region's planners transformed the coastline according to these precepts. With the mass adoption of the automobile in the 1930s and the construction of express freeways in the 1940s and 1950s, urban planners felt an urgent need to turn the beach into an automobile-friendly space. In 1950, the city of Los Angeles built parking spaces for 1,000 cars on the artificiallywidened beach. 18 In 1966, a new express freeway opened that allowed Angelenos to reach the beach from downtown in a just a few minutes. 19 Accommodations were also built to make the trip to the beach more convenient for visitors who came from faraway suburbs and did not want to drive back in their dripping bathing suits: public toilets, showers, bathhouses and brand-new lifeguard headquarters mushroomed along the coast. The buildings were not only functional, they were also meant to be beautiful: according to the SPA Bulletin, a recently-inaugurated modernist building at Will Rogers Beach exemplified "the latest trend in modern beach design." 20 Beach planners also made sure to replace the old amusements parks and dance-halls with "family-friendly" attractions. Pacific Ocean Park (nicknamed "POP"), an amusement park inaugurated in 1958, is a good example of that trend. In the late 1950s, the city of Santa Monica decided that the "dwindling carnival-like attractions" 21 of Ocean Park Pier had run their course. Nostalgic throngs visiting the old pier were not sufficient to sustain the old rides and games. While the pier was preserved, the sideshows gave way to modern attractions. With its ocean theme, POP pretended to be "nothing like Disneyland" but in reality was a very close copy, with similar attractions and a comparable target public: the white middle-class family. Just like Disneyland's racialized landscape of leisure, with attractions such as "Frontierland" featuring colorfully-dressed "savages," POP's most famous rides used stereotypes of non-white cultures to appeal to a white audience This drawing of a building to be built on Will Rogers Beach, a public beach located north of Los Angeles, was published in California Coast in May 1950. With its cubic shape and minimalist design, the building, typical of the 1950s modernist aesthetics, stood in sharp contrast with the old-fashioned beach restaurants and amusement concessions that were built at the beginning of the twentieth century. This type of modern beach buildings was conceived to appeal to the tastes of the middle class. In contrast, the "grimy hot-dog stands" of the past were associated with a workingclass form of leisure. By destroying the latter to make room for the former, urban planners hoped to keep the middle and upper classes on the beach. fascinated by the exotic "Other" (Avila 132-143). By introducing the "Pay-One-Price" single entry-ticket in 1960, POP management also adopted Disneyland's crowd-filtering strategies. 22 Finally, P.O.P. was clean, orderly, and, crucially, it was closely supervised by park attendants in uniform.

Beach Policing in the Atomic Age

Indeed, beach supervision was another crucial development in the postwar transformation of the Los Angeles coastline. Combined with urban renewal techniques that dramatically transformed the beach landscape, 22. Los Angeles Times, April 3, 1960, J3. increased policing contributed to the redefinition of the beach public based on race, class, and-more crucially-gender performance and sexuality. The ideal beach public, according to city officials and the business community, was defined as white. Beach policing in Los Angeles, however, did not specifically target racial minorities. Due to a long history of racial segregation and violence on the sand and the sheer distance between the coastline and most black neighborhoods, the Los Angeles beaches of the 1950s and 1960s were virtually all-white spaces. During this period, the policing of beach visitors was thus not specifically focused on racial minorities but instead on the beachgoers who defied the dominant social and sexual norms. Specifically, the police scrutinized the gay men and lesbians who socialized in beach bathhouses and the "Muscle Beach" bodybuilders who worked out on the sands. In the Cold War era, as historian Elaine Tyler May has explained, any deviation from the norms of appropriate sexual and familial behavior was perceived as a challenge to social order and national security (Tyler May). On the beach, homosexuals and bodybuilders represented a double threat: a social threat-they undermined public decorum and decency-and an economic threat-their presence could affect local real estate value. These threats were linked: if homeowners in communities like Santa Monica or Venice wanted to see real estate values rise, they had to minimize their reputation as a site for Los Angeles bohemia. As a consequence, homosexuals and bodybuilders became the target of several beach policing measures.

Homosexual men and women had gathered on the shores for decades in order to meet potential sexual partners. They became more visible in the postwar era with the opening of several gay bars and restaurants in the old-fashioned bathhouses that the middle-class public had stopped patronizing as they grew comfortable with coming to the beach in their bathing suits or changing in their cars (Faderman & Timmons). Starting in the 1950s, the Santa Monica police was regularly called upon in local newspapers to address the "vice conditions" that existed on the beaches. 23 In 1955 and 1956, two recently-elected city council members, an ordained minister and a lawyer, who had won their seats by campaigning on morality issues, orchestrated a "clean-up" crusade specifically targeting the beach. The first in a long series of anti-gay raids was conducted on a Saturday afternoon of December 1955, when the Santa Monica police and the Los Angeles vice squad stormed an Ocean Park bathhouse and arrested 11 men under the charge of "lewd conduct." Shortly after the raid, the city manager revoked the business license of the bathhouse. But closing is was not enough. To make sure that the so-called "sex deviates" would not return, it was suggested that the bathhouse be razed. 24 This proposal underlines the fact that the beach policing efforts were directly linked with the beach modernization program. When, a year later, another bathhouse was scheduled for destruction following a vice-squad raid, Santa Monica's chief of Police declared that razing the building "w[ould] do much to rid the city of unwelcome sex deviates and other undesirables." 25 But according to Santa Monica's city manager, the bathhouse's destruction "ha [d] been contemplated for over a year as part of the beach master plan." 26 In other words, the decision served both the goals of the moral crusade and the precepts adopted by urban planners. In the days following the second bathhouse raid, it was also suggested to throw more light on the Santa Monica pier's piling and understructure in order to get rid of the "dark corners" where homosexuals were accused of congregating. In both cases, the measures would enhance the beach, "modernize" it, as well as prevent homosexuals from coming to the shores. For their proponents, the conflation of the two programs was logical. Indeed, the beach modernization coalition overlapped with the beach policing advocates: city officials and the mainstream business community supported both endeavors. For them, the voted-upon measures would kill two birds with one stone. The targeting of homosexuals-through anti-gay raids and the eventual destruction of the bathhouses-is one example, among many, of the ways in which beach policing measures collided and reinforced beach modernization efforts. The closing of "Muscle Beach" is another one.

"Muscle Beach" was a section of the Santa Monica beach where acrobats and athletes had gathered since the 1930s. 27 By the 1940s, Muscle Beach had reached national fame drawing packed crowds who watched the acrobatics, body-building contests, and weight-lifting competitions held on the beach stage. Yet in the 1950s the performers started attracting criticism directed at both their bodies and lifestyles, which were seen as atypical. To be sure, Muscle Beach athletes, especially the female and male bodybuilders, boasted unconventional physiques and challenged traditional gender stereotypes. In the newspapers, they were described as "feminine men and masculine women," 28 and this was not a compliment. Moreover, Muscle Beach habitués were perceived as 24. Santa Monica Evening Outlook, December 4, 1955, 1. 25. Santa Monica Evening Outlook, December 4, 1956, 1. 26. Santa Monica Evening Outlook, December 4, 1956, 1. 27. The "Muscle Beach" gym area was originally called the "Santa Monica Beach Playground" and was founded in 1934 by the Work Progress Administration to relieve unemployment (see Evening Outlook, July 16, 1940, 5). The name changed to "Muscle Beach" around the late 1930s-early 1940s. Outlook, December 18, 1958, 4. "drifters" or lazy "beach bums" who did not conform to the dominant model of the male breadwinner and the female homemaker. The exhibition of these new ideals and values did not sit well with the city's authorities. Throughout the 1950s, Santa Monica's conservative citizens, led by councilwoman Alys Drobnick, fought a sustained battle against the muscle shows. But Muscle Beach was popular with the public: Drobnick and her supporters could not fight it only on moral grounds. While they persisted in accusing the performers of being "exhibitionists," they also couched their complaints in terms taken from the beach modernizers' handbook: they accused the site of not being sufficiently supervised and of lacking proper equipment. 29 According to Leonard Bright, the Santa Monica recreation director and an ardent Muscle Beach opponent, the site had to be destroyed and rebuilt in order to foster more "public participation" and less showmanship. 30 In the late 1950s, the area was included in the redevelopment program of Santa Monica's beaches. Yet it was unclear whether the popular Muscle Beach would ever be closed and, if so, when.

Santa Monica Evening

In December 1958, the police arrested five Muscle Beach habitués following police investigation of reports by two young African American runaway girls of 12 and 14. 31 Four of the bodybuilders were accused of sexual misconduct and another man was jailed on misdemeanor charges involving the possession of photographs of male nudity. The press did not explicitly comment on the girls' race but the taboo of interracial sexual relations in Cold War America, even in a city such as Los Angeles where no laws barred interracial marriages, represented an additional cause of concern for city officials who wanted to preserve Santa Monica's reputation as a safe destination for middle-class white families. The 1958 "sex scandal," as it was quickly dubbed by the local press, presented the anti-Muscle Beach coalition with the perfect opportunity to stigmatize Muscle Beach as an unsafe area that should not be present on the beach. Even though the incident had not taken place on the beach-according to the victims' testimonies the crimes had taken place at a beach-front apartment and a Pico Boulevard motel-and though it incriminated only a handful of the Muscle Beach regulars, Drobnick and her allies accused Muscle Beach performers of being child molesters who were partaking in "sex orgies." 32 Their strategy worked: a few days after the arrest, the City Council decided to close the area. It was later filled with dirt to prevent any attempt to restore it and the city declared that a parking lot would be built on the site. Later, it was announced that an "improved Muscle Beach" would be 29. Santa Monica Evening Outlook, December 11, 1958, 1. 30. Santa Monica Evening Outlook, August 18, 1957, 15. 31. Santa Monica Evening Outlook, December 10, 1958, 1. 32. Santa Monica Evening Outlook, December 11, 1958, 1. opened several hundred feet to the south of the existing weightlifting platform, effectively shifting the locus of control over the beach from the bodybuilders to the city.

Muscle Beach regulars protested the closing of their gym. Many wrote letters to the local newspapers to protest their characterization as "undesirable characters." One letter claimed that Muscle Beach participants included "doctors, lawyers, police officers, actors, etc." 33 Some of them even blamed the "'homo' carnival that prance [d] along 'Santa Monica promenade'" 34 for tarnishing their reputation and demanded more policing around the platform. But the area remained closed. In the words of Santa Monica's police chief, Muscle Beach had created a "terrific sex deviate problem" 35 on the beach, attracting homosexuals and child predators. Muscle Beach's demise was therefore part of the larger moral crusade fought against any beachgoer who did not conform to the prevalent sexual and gender norms. While gay men and lesbians were more specifically targeted by these measures, they also affected all the performers who, by their choice of clothing or of athletic activity, had threatened the status quo. For the anti-Muscle Beach coalition, women who lifted weights and men who wore tight bathing suits while showing off their muscles had no place on the sands. According to an anti-Muscle Beach petition signed in 1959 by several concerned citizens, the beaches were deemed reserved "for the recreational use of healthy-minded young people, families with children and older citizens." 36 The "proper" beach public could not be more clearly defined. According to this vision, the beach was the exclusive recreational territory of the nuclear family and the elderly.

The construction of the "new" muscle beach gym area confirmed the city's intentions regarding the type of public it wished to attract. The recreation commission recommended the opening of a "family-style outdoor gym area," "under strict municipal supervision." 37 Indeed at the "new" Muscle Beach that opened in August 1959, three supervisors were on duty and, as underlined by Leonard Bright, they had the power to make arrests. 38 In order to discourage any form of "exhibitionism," the gym area did not include a platform. Moreover, the new Muscle Beach, which basically resembled a children's playground, differed from the older one in its strict segregation of children's and adults' equipment. In other words, the new site fulfilled the main goal of 33. Santa Monica Evening Outlook, December 16, 1958, 4. 34. Santa Monica Evening Outlook, December 16, 1958, 4. 35. Santa Monica Evening Outlook, December 17, 1958, 1. 36. Santa Monica Evening Outlook, June 12, 1959, 1. 37. Santa Monica Evening Outlook, December 17, 1958 ; Santa Monica Evening Outlook, December 11, 1958, 1. 38. Santa Monica Evening Outlook, August 10, 1959, 11. Photographs commissioned by the Los Angeles Daily News, Los Angeles Daily News Negatives, Department of Special Collections, Charles E. Young Research Library, University of California at Los Angeles.

On this photograph, a young child is hoisted to the top of a human pyramid at Muscle Beach (1947). Acrobats, both young and old, were very popular with the crowds. Moreover, because they did not challenge the dominant gender and bodily norms, they did not attract the harsh criticism that targeted male and female bodybuilders. Yet the acrobats were also chased from the beach following the 1958 "sex scandal" that resulted in Muscle Beach's closing. Indeed the informal atmosphere of Muscle Beach and its lack of supervision did not fit the vision of the modern beach developed by local officials and urban planners. With its unruly and intergenerational crowds huddled around a decrepit wooden platform, Muscle beach belonged to the past.

the anti-Muscle Beach coalition: it would prevent the informal organization of muscle shows while ensuring that a respectable "family atmosphere" would reign on the beach. 39 Meanwhile, the beach modernization program contributed to the destruction of the infamous muscle Beach "community" that had emerged with the shows. By razing the decrepit apartments and hotels along the beach where many of the stalwarts used to live, the city was not only improving the beaches: it was ensuring that the athletes would not return. Beach modernization once more served the goals of beach policing, while police surveillance of the beaches cleared the way for the modernizers' plans. Times, August 9, 1959, WS1; Santa Monica Evening Outlook, August 10, 11 While these measures did deter the athletes from coming back to Santa Monica, most of them found their way back to the shore. But this time, they chose a different beach: Venice. In the 1950s, the neighborhood had turned into "the 'third' beat community" (Maynard 13), attracting hundreds of artists and dropouts who sought cheap rents and the sea breeze. Although less famous than Greenwich Village in New York and North Beach in San Francisco, the declining resort had a reputation as a Bohemian paradise where it was possible to live a counterculture lifestyle. This was no surprise, then, that the Muscle Beach athletes chose Venice as their new home. Venice already had its own weight-lifting platform, which had been inaugurated in 1950 during the height of the beach modernization era. But it had remained under the shadow of its famous predecessor in Santa Monica. In the 1960s, however, the Venice weight-lifting pit developed its own following thanks, in large part, to the influx of Muscle Beach refugees. By the 1970s, Arnold Schwarzenegger's multiple bodybuilding titles and his first movie success in Pumping Iron (G. Butler and R. Fiore, 1977) had drawn attention to the gym area where he had started his training as a young Austrian immigrant. Described as a "debauched Disneyland" by a journalist, "Muscle Beach Venice," as it had been renamed by the habitués, fit in well with the neighborhood. As the region's "last poor beach," Venice was the final rampart on the coast where hippies, streets musicians, and peddlers were welcomed. 40 Everywhere else on Los Angeles's shoreline, urban renewal and increased policing had done their job: the socalled "undesirables" were nowhere to be seen. *** By the early 1960s, Los Angeles finally had the beaches it deserved as a modern coastal metropolis: clean, wide, and equipped with modern facilities. More crucially, by "cleaning up" the beaches, both literally-when they destroyed the hot-dog stands and amusement concessions that had long been the favorite part of the beach experience for working-class visitors-and figuratively-when they conducted anti-gay raids and closed down Muscle Beach-the members of the beach modernization coalition and the advocates of beach policing cleared the way for those they perceived as deserving of publicly-funded recreational spaces: middle-and upper-class white families and, to some extent, the elderly. While the city's public beaches remained opened to everybody, at least in theory, they were imagined, planned for, and adapted to this specific public in mind.

Los Angeles

The strategies deployed by urban planners, business elites and local politicians did work: unlike Coney Island and Atlantic City, the beaches of Los Angeles remained attractive to the white middle class. But there were 40. Los Angeles Times, July 11, 1981, OC-C1; Los Angeles Times, November 5, 1967, B20. also many setbacks: as mentioned above, Venice Beach remained a favorite haunt for "everything that [wa]s perceived as wrong or weird in Southern California" 41 . Gay men and lesbians, as described in John Rechy's semiautobiographical novel City of Night (1963), did come back to Santa Monica, some of them in plain view [START_REF]See in particular his description of Crystal Beach, the gay stretch of the beach in Santa Monica[END_REF] . Moreover, the 1970s brought new obstacles for the proponents of beach modernization and policing. In Santa Monica, a grassroots campaign was carried out in 1973 in order to save the municipal pier from its scheduled destruction. According to many, it was precisely because the pier "lured a deliciously rich cross section of people and characters" 43 that it had to be preserved. The pier remained intact. Meanwhile, in the years following the 1965 immigration reform, the ethnic and racial diversity of the beach public dramatically expanded, challenging the informal segregation that had persisted for decades. In 1972, Santa Monica lifeguards mournfully compared the beach to a "melting-pot at the end of a freeway" (Edgerton 51). By the mid-1970s, the consensus that had brought together urban planners, businessmen, and local officials on what should be built on the beach and who should have access to it started to crack. The influx of Latino and Asian immigrants in the city, the rising number of critics condemning urban renewal and advocating for the need to preserve historic buildings, and the growing assertiveness of gay men and women in public spaces are some of the forces that challenged the status quo. But the collapse of the coalition that reigned over the coastline from the 1940s until the late 1960s did not put an end to the tensions and debates concerning who has "the right to the beach." New strategies of exclusion have appeared while traditional methods used to filter the crowd persist. Most recently, owners of private beach houses in Malibu have used fake "no parking" signs to prevent the general public from coming to the shore. The "right to the beach" in Los Angeles was, and remains, contested. While Angelenos need to be vigilant to prevent the rampant privatization of their public beaches by coastline homeowners, they should also be aware-as this article has demonstratedthat the ways in which a beach is planned, developed, and policed has a major impact on who will come to, and feel comfortable on, the sand.

  10. "Report on the Revised Master Plan of Shoreline Development." County of Los Angeles, 1946, Southern Regional Library Facilities, UCLA; "Recreational Development of the Los Angeles Area Shoreline: An Engineering and Economic Report to the Mayor and the City Council, City of Los Angeles," New York, 1949, 148, Box B1380, LACA. California Coast 3:3 (August 1949): 3. © Doheny Memorial Library, University of Southern California. This picture, entitled "Not Coney Island-but Santa Monica on a Hot Sunday," was published in the official newsletter of the Shoreline Planning Association, an organization dedicated to the preservation and development of the California's shoreline in August 1949. The comparison with Coney Island-the Eastern beach famous for its large crowds-and the photograph showing teeming crowds of teenagers huddled together on Santa Monica Beach were meant to alert the state's inhabitants and officials to the dire situation of the beaches in the postwar period.

  13. Western City, April 1950, 28. 14. "Historical Changes in the Beaches of Los Angeles County: Malaga Cove to Topanga Canyon: 1935-1990," Department of Beaches and Harbors, County of Los Angeles, 1992, Santa Monica Public Library. California Coast 5:2 (July 1948): 3. © Doheny Memorial Library, University of Southern California.

15.

  California Coast 4:2 (May 1950): 2. 16. On Disneyland and "disneyfication," see Sharon Zukin, Landscapes of Power (1991); Michael Sorkin, ed., Variations on a Theme Park (1992); Eric Avila, Popular Culture in the Age of White Flight (2004).

  A. Scott, Santa Monica: a History on the Edge (Mount Pleasant, SC: Arcadia Publishing, 2004) 132. 20. California Coast 4:2 (May 1950): 3. 21. LA Examiner, January 10, 1957. California Coast 2:4 (May 1950): 2. © Doheny Memorial Library, University of Southern California.

In its composition, the beach modernization group resembled the "pro-growth coalitions" described by John Mellonkopf. According to him, these coalitions, which found their origins in Democratic politics, united public and private interests behind government programs (in this case the 1949 Housing Act, which allowed cities to obtain federal funding for urban renewal projects). Mellonkopf argues that these coalitions were instrumental in bringing about the dramatic transformations of postwar American cities(Mollenkopf).

12. "A Sanitary Survey of Sewage Pollution of the Surf and Beaches of Santa Monica Bay," 1942, Box A826. File 14669, ACLA.

Santa Monica Evening Outlook, April 5, 1955, 1; Santa Monica Evening Outlook, April 2, 1955, 1; Santa Monica Evening Outlook, December 5, 1955, 1.


Friedrich Wagemann

Deformations of Lie algebras of vector fields arising from families of schemes

Introduction

The goal of the present paper is to construct examples of global deformations of vector field Lie algebras in a conceptual way. Fialowski and Schlichermaier [START_REF] Fialowski | Global Deformations of the Witt algebra of Krichever-Novikov type[END_REF] constructed global deformations of the infinitesimally and formally rigid Lie algebra of polynomial vector fields on the circle and of its central extension, the Virasoro algebra. The Lie algebra of polynomial vector fields on the circle is here replaced by/seen as the Lie algebra of meromorphic or rational vector fields on the Riemann sphere admitting poles only in the points 0 and ∞. In this context one gets non trivial deformations from an affine family of curves by first deforming it as a projective family with marked points and then extracting the points. Fialowski and Schlichermaier get in this way non-trivial deformations of Lie algebras, and the underlying families of curves present singularities.

In the attempt of producing deformations of vector field Lie algebras from deformations of the underlying pointed algebraic variety in a general framework, we are led to a notion of global deformation which is different from the one used by Fialowski and Schlichenmaier, one which is closer to deformations in algebraic geometry. A first goal is to compare these global deformations with the corresponding notion from Fialowski-Schlichermaier's article.

We then stick to the notion of deformations of the Lie algebra of vector fields on a pointed algebraic curve imposed by the deformation of the underlying curve, and show its close relation to the moduli space of pointed curves. In order to formulate this relation, we show that the "space of deformations" carries the geometric structure of a C-stack. This is a kind of functor from the category of model spaces, here affine schemes, to the category of groupoids (in order to capture the idea of considering deformations up to isomorphism). It has to satisfy conditions to provide construction of geometric objects by gluing local data, making it ressemble the functor of points of a scheme. The link between the moduli stack M g,n and the deformation stack Def is a morphism of stacks I. It realizes a family of marked projective curves as a deformation of the Lie algebra of regular vector fields on the affine curve obtained from extracting the points, by extracting the divisor associated to the union of the marked points, and then taking the Lie algebra of sections of the relative tangent bundles of the family.

We show furthermore that I is almost a monomorphism. The idea that this might be true comes from Pursell-Shanks' theory of describing the underlying manifold (and more geometric objects related to it) by its Lie algebra of tangent vector fields. We generalize this theory to a relative (affine) setting, i.e. we show that over C, Der A (B) ∼ = Der A (B ′ ) implies an A-isomorphism B ∼ = B ′ .

Let us observe that all which is discussed in this paper can be easily transposed to current algebras, instead of vector field Lie algebras. We hope that the stack approach to deformations is useful from the two points of view: our believe is that on the one hand it may give rise to a cohomology theory gouverning global deformations by associating some standard homological algebra tool to the above morphism of stacks I (its cotangent complex ?). For this direction, it might be necessary to show first that some substack of our deformation stack is an algebraic stack, cf the remark at the end of section 5. Acknowledgements: the author thanks Alice Fialowski for the discussion at Strasbourgh which initiated the present study. He thanks furthermore Pierre-Emmanuel Chaput for help with lemma 3 and other useful remarks, as well as Matthias Borer for useful discussions. Some of this work was discussed with Martin Schlichenmaier at Luxembourg, and my thanks go to him as well.

Global deformations of general and of product type

Let C be the base field; we are conscious that some results are valid in a much more general framework.

In this first section, we compare two types of global deformations of Lie algebras. On the one hand, we have global deformations as Fialowski-Schlichenmaier [START_REF] Fialowski | Global Deformations of the Witt algebra of Krichever-Novikov type[END_REF] define them. We will call these of product type. In the following, A is an associative commutative unital C-algebra (and all C-algebras will be supposed associative commutative and unital).

Definition 1 For a Lie algebra g 0 , a global deformation (of product type) with base A is an A-Lie algebra structure [-, -] λ on the tensor product g 0 ⊗ C A, together with a morphism of Lie algebras id ⊗ ǫ : g 0 ⊗ A → g 0 ⊗ C ≃ g 0 , where ǫ is the augmentation of the (augmented) algebra A.

Concretely, the bracket [-, -] λ is antisymmetric, satisfies the Jacobi identity, and the following two identities for a, b ∈ A and x, y ∈ g 0 owing to A-linearity of the bracket and the fact that ǫ ⊗ id is supposed to be a Lie algebra homomorphism:

(1) [x ⊗ a, y ⊗ b] λ = [x ⊗ 1, y ⊗ 1] λ (ab) (2) (id ⊗ ǫ)([x ⊗ 1, y ⊗ 1] λ ) = [x, y] ⊗ 1.
2 By condition [START_REF] Altman | Introduction to Grothendieck Duality Theory[END_REF], it is enough to give the commutators [x ⊗ 1, y ⊗ 1] λ for all x, y ∈ g 0 in order to describe the deformation. By condition [START_REF] Amemiya | Lie algebra of vector fields and complex structures[END_REF], their bracket has the form

[x ⊗ 1, y ⊗ 1] λ = [x ⊗ 1, y ⊗ 1] + i z i ⊗ a i .
We call the notion with which we want to contrast these global deformations of product type general global deformations. We take over the term "global" from earlier work on the subject, notwithstanding the fact that these deformations are "only" on affine open sets (namely, on the affine open set given by Spec(A), the spectrum of A).

Definition 2 Let g 0 be a C-Lie algebra. A general global deformation of g 0 is an A-Lie algebra g together with an augmentation ǫ : A → C such that g ⊗ A C ≃ g 0 as C-Lie algebras (where A acts on C via ǫ).

One detail is hidden in this definition. g ⊗ A C is a C-Lie algebra: indeed, g ⊗ A C is a C-Lie algebra with the current algebra bracket [x ⊗ α, y ⊗ β] := [x, y] ⊗ αβ which we call the trivial bracket here. Then the space

I := span C {ax ⊗ α -x ⊗ ǫ(a)α | x ∈ g 0 , a ∈ A, α ∈ C }
is a Lie-ideal as follows directly from the A-linearity of the bracket on g, so the quotient g ⊗ A C is a Lie algebra.

Proposition 1 A deformation of product type (g 0 ⊗ C A, [, ] λ , ǫ) is a general global deformation.
Proof . It is clear that (g 0 ⊗ C A) ⊗ A C ≃ g 0 as C-modules. The morphisms can be given by

f : (g 0 ⊗ C A) ⊗ A C → g 0 , f (x ⊗ a ⊗ α) = αǫ(a)x = α(id ⊗ ǫ)(x ⊗ a) and g : g 0 → (g 0 ⊗ C A) ⊗ A C, g(x) = x ⊗ 1 ⊗ 1.
Using that the A-module structure on C is given by the augmentation ǫ, it is easy to see that f and g are mutually inverse. Let us show that f is a morphism of Lie algebras:

f ([x ⊗ a ⊗ α, y ⊗ b ⊗ β]) = f ([x ⊗ a, y ⊗ b] λ ⊗ αβ) = f ([x ⊗ 1, y ⊗ 1] λ (ab) ⊗ αβ) = f ([x ⊗ 1, y ⊗ 1] λ ⊗ ǫ(ab)αβ) = ǫ(ab)(αβ) (id ⊗ ǫ)([x ⊗ 1, y ⊗ 1] λ ) = ǫ(ab)(αβ) [x, y].
where we used conditions (1) and ( 2), and on the other hand

[f (x ⊗ a ⊗ α), f (y ⊗ b ⊗ β)] = ǫ(ab)αβ [x, y].
This ends the proof of the lemma.

Thus, the notion of general global deformations includes the deformations of product type, as it should. The reason for considering these more general deformations is that in general the Lie algebra deformations of a Lie algebra of vector fields coming from deformations of the underlying variety are not of product type.

A map from families of schemes to deformations of Lie algebras

A(n affine) deformation of a scheme X 0 over C is a scheme X over a C-algebra A and a closed point 0 ∈ Spec(A) (with residue field C) such that the fiber (X) 0 of X in 0 satisfies (X) 0 ≃ X 0 as C-schemes. Let us denote by π : X → Spec(A) the structure morphism. Usually we suppose furthermore that π : X → Spec(A) is flat, quasi-compact, surjective and finitely presented. Recall that when π is of finite type and A noetherian, π is automatically of finite presentation. As one encounters non-noetherian bases in descent theory, finite presentation is the good hypothesis to impose (for example, in the realm of fppf topology). We then regard X as a family of schemes deforming X 0 . By definition of the fiber, (X) 0 ≃ X 0 means X × Spec(A) Spec(C) ≃ X 0 .

In order to get started, let us suppose that the family is affine, i.e. X = Spec(B), and that X 0 is affine also, i.e. X 0 = Spec(B 0 ). We are interested in the Lie algebra of vector fields on X which are tangent to the fibers of π, i.e. in the Lie algebra of A-linear derivations of B, denoted Der A (B).

The following lemma expresses the behavior of A-linear derivations of B under base change.

Lemma 1 Let A be a C-algebra, and A ′ and B be A-algebras. Let B ′ := B⊗ A A ′ the coproduct in this situation. Suppose that B ′ is a flat B-module and that the B-module of Kähler differentials Ω A (B) has a finite presentation. Then

Der A ′ (B ′ ) ≃ Der A (B) ⊗ A A ′ as A ′ -modules.
Remark 1 Translating into scheme language, i.e. X = Spec(A),

X ′ = Spec(A ′ ), Y = Spec(B), Y ′ = Spec(B ′ ) is the fibered product Y ′ = X ′ × X Y ,
and the hypotheses are satisfied in case the induced morphisms π : Y → X and X ′ → X are of finite presentation and flat. This situation is called base change (in the fppf topology).

Proof . This follows in a straight forward manner from the corresponding base change for the modules of Kähler differentials. Here is an explicit proof:

The A ′ -module of Kähler differentials Ω B ′ /A ′ of the A ′ -algebra B ′ represents the functor Der A ′ (B ′ , -) which implies in particular an isomorphism of B ′modules

Der A ′ (B ′ ) ≃ Hom B ′ (Ω B ′ /A ′ , B ′ ).

The hypothesis B ⊗

A A ′ = B ′ implies Ω B ′ /A ′ ≃ Ω B/A ⊗ B B ′ , which in turn gives Hom B ′ (Ω B ′ /A ′ , B ′ ) ≃ Hom B ′ (Ω B/A ⊗ B B ′ , B ′ ).
Then we apply the "change of rings" isomorphism (meaning the adjointness of -⊗ B B ′ and Hom B ′ (B ′ , -))

Hom B ′ (Ω B/A ⊗ B B ′ , B ′ ) ≃ Hom B (Ω B/A , B ′ ).
In order to retranslate the result into Der A (B) ⊗ B B ′ , we need for a B-module M a natural isomorphism

Hom B (M, B ′ ) ≃ Hom B (M, B) ⊗ B B ′ .
There is obviously such an isomorphism for M = B. Furthermore, the claim holds true for a B-module M of finite presentation as is easily checked (B ′ is a flat B-module by hypothesis). But Ω B/A has a finite B-presentation (here we also use the noetherian hypothesis). Thus we can conclude

Hom B (Ω B/A , B ′ ) ≃ Hom B (Ω B/A , B) ⊗ B B ′ ,
and

Hom B (Ω B/A , B) ⊗ B B ′ ≃ Der A (B) ⊗ B B ′ . Finally Der A (B) ⊗ B B ′ = Der A (B) ⊗ B (B ⊗ A A ′ ) ≃ Der A (B) ⊗ A A ′ .
Remark 2 It is easy to check that the above isomorphism

Der A ′ (B ′ ) ≃ Der A (B) ⊗ A A ′
is an isomorphism of A ′ -Lie algebras when we give the right hand side the bracket [x ⊗ a, y ⊗ b] := [x, y] ⊗ ab. This follows from the fact that the bracket of derivations D, D ′ is on both sides given by [D,

D ′ ] = D • D ′ -D ′ • D.
Lemma 2 Let A be a C-algebra, B → A be an A-algebra, and A → C be a C-point of the affine scheme Spec(A). Suppose that the B-module Ω B/A is projective. Then there is an isomorphism of C-modules

Der A (B) ⊗ A C ≃ Der C (B 0 ).
Proof . This is a slightly different situation from that in lemma 3 where the module of derivations of the fiber (product) is just the tensor product of the initial module of derivations. Its justification stems from the fact that in the above proof of lemma 3, the natural isomorphism

Hom B (M, B ′ ) ≃ Hom B (M, B) ⊗ B B ′
holds not only for M = B n , but also for a direct factor of B n , i.e. for each projective B-module, by additivity of the concerned functors. Observe that we do not need any flatness assumption here.

Remark 3 Translated into scheme language, this lemma shows that the Lie algebra of derivations of a fiber of π : Spec(B) → Spec(A) is the tensor product of Der A (B) by C, seen as an A-module via the augmentation A → C, in case A is smooth. More generally, this is true at each smooth point of A.

Translated into the language of deformations exposed in section 1, it shows that deformations of the underlying affine scheme X 0 = Spec(B 0 ) give rise to general global deformations of the Lie algebra of derivations Der(B 0 ).

In conclusion, there is a map, which we call I, assigning to an affine family of schemes X (over A) deforming a given affine scheme X 0 a general global A-deformation of the Lie algebra V ect(X 0 ). Remark 4 Note that the map I does not define a functor in general, as Der A (B) is not functorial in B; the point is that pushforward or pullback of vector fields along morphisms works for isomorphisms.

For example, one can define a genuine functor when restricting the source category to a groupoid of some kind of varieties with allowed morphisms being only the isomorphisms.

Corollary 1 In the case of a product Y ′ = Y × X ′ and X = Spec(C), i.e. a trivial deformation, one gets an isomorphism of Lie algebras

Der A ′ (B ′ ) ≃ Der C (B) ⊗ C A ′ .
Remark 5 Here the right hand side carries the current algebra bracket [x ⊗ a, y ⊗ b] := [x, y] ⊗ ab which is considered as the trivial deformation, thus we see that trivial deformations of the underlying variety give rise to trivial global deformations of product type.

Remark 6

The difference between the notions of general global deformations and of global deformations of product type is that the latter considers only global deformations which are, as A-modules, tensor products with A over C. But these are then free A-modules, whereas in general the Lie algebra of A-linear derivations on an algebra B has no reason to be a free A-module. It would be interesting to have a clear criterion when this is the case.

Nevertheless, in case the special point 0 ∈ Spec(A) is a generic point in the moduli space M g,n , Der A (B) is a free A-module, because in this situation, the sheaf of Krichever-Novikov algebras is locally free, and the local basis trivializes the module, see [START_REF] Schlichenmaier | Knizhnik-Zalmolodchikov equations for positive genus and Krichever-Novikov algebras[END_REF] p. 743. We owe this remark to M. Schlichenmaier.

Extracting affine families from proper families

Let S be some scheme over C and n, g integers such that 2g -2 + n > 0. A proper flat (surjective) family π : C → S whose geometric fibers C s are reduced connected curves of genus g = dim H 1 (C s , O Cs ) with at most ordinary double points is called a stable curve of genus g over S. An n-pointed stable curve is a stable curve of genus g over S together with n sections σ 1 , . . . , σ n : S → C of π such that the images P i = σ i (s) are disjoint and smooth for all i = 1, . . . , n and all s ∈ S. Furthermore, the number of points where a nonsingular rational component E of C s meets the rest of C s plus the number of points P i on E must be at least 3, cf [START_REF] Finn | The projectivity of the moduli space of stable curves II: the stacks M g[END_REF] definition 1.1 p.162. In the same vein, we call an affine stable curve a flat (surjective) family π : C → S whose geometric fibers are affine stable reduced and connected curves of genus g with at most ordinary double points. All our affine stable curves will be obtained from n-pointed stable curves by extracting the marked points and one may take this for the definition of an affine stable curve. We will then be in position to apply the results of section 2: Lemma 3 Let π : C → S be a punctured stable curve, and σ 1 , . . . , σ n the corresponding sections of π. Denote by △ the union △ = n i=1 σ i (S). Then π : C \ △ → S is an affine stable curve and an affine morphism.

Proof . It is shown in [START_REF] Finn | The projectivity of the moduli space of stable curves II: the stacks M g[END_REF] p. 173 corollary 1.9 that some tensor power of the sheaf ω Cs/C (P 1 + . . . + P n ) is very ample, where ω Cs/C is the dualizing sheaf of C s over C. Here we denote by C s = π -1 (s) for s ∈ S, and P i = σ i (s) for i = 1, . . . , n. Let us denote by L △ the line bundle ω C/S (△) on C. By Grauert's theorem (cf [START_REF] Hartshorne | Algebraic Geometry[END_REF] pp. 288-289), we have R i π * (L ⊗n △ ) = 0 for all i > 0 and n >> 0. The local to global spectral sequence gives therefore H i (C, L ⊗n △ ) = 0 for all i > 0 and n >> 0. L △ is then an ample line bundle on C, and we use some power L ⊗k △ of L △ to embed C into projective space P N A ; let i : C → P N A be corresponding embedding. Then C \ △ maps under i into the complement of a hyperplane, and i| C\△ is affine. The claim follows. Now we pass to moduli space; there are (at least) two points of view on the moduli space of curves, and it is the necessity of the situation which determines which one to take. One point of view is the following (the GIT or coarse moduli space point of view): the moduli space (of stable connected projective algebraic curves over C of fixed genus g with n marked points) is a scheme whose geometric points represent the isomorphism classes of stable punctured curves. We observe in lemma 4 below that isomorphic punctured curves give rise to isomorphic deformations, and get thus that I factors to a map from the set of S-points of the moduli space to global deformations of vector field Lie algebras over S.

More precisely, let π : C → S and π ′ : C ′ → S be two isomorphic punctured curves by an S-morphism φ which respects the punctures. Let U = Spec(B) ⊂ C be an affine open set such that π| U and π| φ(U) are affine families of schemes. Denote by B and B ′ the corresponding algebras, i.e. U = Spec(B) and φ(U ) = Spec(B ′ ). One may assume that π(U ) = π ′ (φ(U )) = Spec(A).

Lemma 4 In the above notation, the isomorphic curves π : C → S and π ′ : C ′ → S give rise to isomorphic Lie algebras Der A (B) and Der A (B ′ ).

On the other hand, let π : C → S and π ′ : C ′ → S be two punctured curves, let T = Spec(A) ⊂ S be an affine open subset of the parameter space and denote by U = Spec(B) and

U ′ = Spec(B ′ ) the affine open sets U = π -1 (T ) and U ′ = (π ′ ) -1 (T ) corresponding to the △ (resp. △ ′ ) complement in C (resp. C ′ ).
Theorem 1 In the above notation, suppose

• either that B and B ′ are integral, normal, of finite type over A noetherian,

• or that all fibers of C and C ′ are smooth curves,

• or that Der A (B) (resp. Der A (B ′ )) is a free B (resp. B ′ ) module of rank 1.
Then Der A (B) ≃ Der A (B ′ ) as Lie algebras over A implies that there exists a T -isomorphism φ : U → U ′ which is compatible with the punctures.

Remark 7

The special case where T is a closed point of residue field C is easily solved: the claim is then the fact that an isomorphism between punctured projective curves f : (X \ {p 1 , . . . , p n }) ≃ (Y \ {q 1 , . . . , q n }) can be extended to an isomorphism X ≃ Y respecting the punctures (use proposition 6.8 in [START_REF] Hartshorne | Algebraic Geometry[END_REF] p. 43 and induction applied to i

• f : (X \ {p 1 , . . . , p n }) → (Y \ {q 1 , . . . , q n }) ֒→ Y ).
The proof of the above theorem follows from Pursell-Shanks theory, and is given in the next section.

One deduces from the theorem that we get a well-defined injective map from the set of S-points of the moduli space of punctured smooth curves (deforming a fixed punctured curve (X 0 , p 1 , . . . , p n )) to the set of isomorphism classes of deformations of the Lie algebra V ect(X 0 \ {p 1 , . . . , p n }) with base S.

Pursell-Shanks theory

In this section, we develop another key ingredient of our study of the moduli space of deformations of a given Lie algebra of regular vector fields on an affine curve. Here we recall how the Lie algebra of vector fields on a manifold encodes geometric objects of the manifold. This theory is due to Pursell-Shanks, Omori, Amemiya [START_REF] Amemiya | Lie algebra of vector fields and complex structures[END_REF], Grabowski [START_REF] Grabowski | Isomorphisms and Ideals of the Lie algebra of Vector Fields[END_REF], and in our framework to Siebert [START_REF] Siebert | Lie algebras of derivations and affine algebraic geometry over fields of characteristic 0[END_REF].

To give an example of the kind of proposition studied in this theory, Siebert proves that two complex normal reduced irreducible affine algebraic varieties Y = Spec(B) and Y ′ = Spec(B ′ ) over C are isomorphic if and only if their Lie algebras of vector fields are isomorphic as Lie algebras (over C). The method of proof describes the points of the underlying variety in terms of data attached to the Lie algebra of vector fields L. Indeed, he proves for irreducible Y a bijection between the regular points of Y and the finite-codimensional maximal subalgebras L of L such that L does not contain a proper Lie ideal. Call the set of these subalgebras M. The idea is that such an L is the Lie subalgebra of vector fields vanishing at the point it represents. This bijection can be used to express the Zariski topology on Y in Lie algebraic terms. Then Siebert defines an algebra of functions B(L) (cf definition 10 [20]) on Y by taking those functions f : M → C such that for all δ ∈ L, there is a θ ∈ L such that f (L)δ -θ ∈ L for all L ∈ M. B(L) encodes thus the coefficient functions of the vector fields. Our idea is that the above theory reflects that the map I is injective in a certain sense. For this we need a relative version of the above statement, namely Der A (B) ≃ Der A (B ′ ) if and only if Spec(B) ≃ Spec(B ′ ) by an isomorphism over Spec(A). The following statement will be the main result of this section.

Proposition 2 Suppose

• either that B and B ′ are integral, normal, of finite type over A noetherian,

• or that all fibers of C and C ′ are smooth (cf notations section 3),

• or that Der A (B) (resp. Der A (B ′ )) is a free B (resp. B ′ ) module of rank 1.

Then Der A (B) ≃ Der A (B ′ ) if and only if Spec(B) ≃ Spec(B ′ ) by an isomorphism over Spec(A).

Proof . Denote as before Der A (B) = L. It is a modular Lie algebra in the sense of definition 1 of [START_REF] Siebert | Lie algebras of derivations and affine algebraic geometry over fields of characteristic 0[END_REF] (with respect to B). To every algebra homomorphism φ : A → C, i.e. to every C-point x of Spec(A) = X, we associate the fiber π -1 (x) of π : Y = Spec(B) → X. Now we apply Siebert's theory to describe the smooth fibers π -1 (x): φ gives rise to the C-algebra B ⊗ A C where A acts on C via φ and on B via A → B. For each point φ = x ∈ X, there is a projection

π φ : Der A (B) → Der A (B) ⊗ A C → Der A (B ⊗ A C)
given by the composition of X → X ⊗ 1 and

D ⊗ 1 → (b ⊗ 1 → D(b) ⊗ 1).
If all fibers are smooth, B is even more readily identified with B. If L is a free rank 1 B-module with generator δ say, we have for f ∈ B by definition f δ = θ, but on the other hand f ′ δ = θ for some f ′ ∈ B. It follows f = f ′ on all M φ , which permits to conclude in this case.

Remark 8 In the smooth case, one can also argue in several different ways: in case Y = Spec(B), X = Spec(A) and π : Y → X are smooth and T π is surjective, the tangent bundles T Y of Y , and T X of X fit into an exact sequence

0 → T Y /X → T Y → π * T X → 0,
defining the relative tangent bundle T Y /X whose regular sections are Der A (B). One can show that Der A (B) is an admissible B-Lie module in the same way Amemiya showed it for Lie algebras defining pseudo foliations.

One can also show directly that Der A (B) is "strongly nowhere vanishing" in the sense of Grabowski, in the smooth setting.

Let us finish the proof of theorem 1: it remains to show that two families of punctured curves π : C → S and π ′ : C ′ → S with associated divisors △ and △ ′ such that φ : C \ △ ∼ = C ′ \ △ ′ as schemes over Spec(A) are isomorphic as schemes over Spec(A). Indeed, a point where φ is not defined is on some fiber of π (the families are surjective), but there the restriction of φ defines a rational map of curves, thus there is no such point.

The stack of deformations

It is well known that the moduli space of curves carries a structure of an algebraic scheme (the quasi-projective irreducible coarse moduli scheme), or the structure of an algebraic stack. The first approach follows from geometric invariant theory (GIT) and has the advantage that one stays within the framework of standard algebraic geometry. The second approach stems from Grothendieck's program of characterizing the functor of points h X : Sch → Sets, S → Hom Sch (S, X) associated to a fixed scheme X within all functors F : Sch → Sets. Some necessary conditions generalized from this example lead to the definition of a C-stack, a device which generalizes schemes and which is important in the study of classification problems in algebraic geometry where the "classifying space" is not a scheme any more. Indeed, in the category of (algebraic) stacks, the moduli functor is representable, thus it does not only posess a coarse, but a fine moduli stack. The disadvantage of the stack approach is that one has to (re)learn how to do (algebraic) geometry with stacks instead of schemes, varieties or manifolds. One advantage of the stack approach is that it includes (at least formally) the approach by coarse moduli spaces, cf rem. (3.19) [START_REF] Laumon | Champs algébriques[END_REF]. The GIT is used in the context of stacks in order to show that some stacks are algebraic.

Our understanding of the notion of algebraic or differentiable stacks is based on [START_REF] Grothendieck | SGA 1: Revêtements étales et Groupe Fondamental[END_REF], [START_REF] Metzler | Differentiable stacks[END_REF], [START_REF] Vistoli | Notes on Grothendieck topologies, fibered categories and descent theory[END_REF].

When we address the question about some kind of structure of a variety on the space of deformations, it will lead naturally to the question whether the map I respects these structures.

Let us define a stack of deformations of Lie algebras. It is some kind of functor, more precisely a lax functor, which does not associate to an affine scheme Spec(A) the set of isomorphism classes of A-Lie algebras, but rather the groupoid of A-Lie algebras, keeping the data of the iso-and automorphisms between/of A-Lie algebras. There are (at least) two points of view on stacks; let us use here the one based on lax functors (or 2-functors).

As in any geometric theory, one needs first to fix a class of standard spaces to which the spaces one wants to define, should be locally isomorphic. Led by the example of the category of open sets of a manifold (the ones isomorphic to R n serving exactly the expressed need), one defines a Grothendieck (pre-)topology on a category C to be a collection of covering families T (U ) for each object U of C such that covering families contain isomorphisms, are stable under base change, and are stable under refinement. The category of interest for us will be (Aff/C), the category of affine schemes over C, and as Grothendieck topologies on (Aff/C), we will regard the classical four topologies fpqc, fppf, étale and Zariski. The preceding abbreviations mean fidèlement plat, quasi-compact, and fidèlement plat, de présentation finie respectivement. This indicates the conditions on a morphisms between affine schemes to be a member of the covering family. We refer to [START_REF] Altman | Introduction to Grothendieck Duality Theory[END_REF] for basics about faithfully flat or quasicompact morphisms, or morphisms of finite presentation. The four topologies are ordered from finest (fpqc) to coarsest (Zariski). See also the precisions made by Kleiman and Vistoli on the fpqc topology in [START_REF] Vistoli | Notes on Grothendieck topologies, fibered categories and descent theory[END_REF].

A lax functor associates to an affine scheme a groupoid. This association is not an ordinary functor, but a 2-functor. Here the category Aff/C is seen in a trivial way as a 2-category (the 2-morphisms are identities between compositions of 1-morphisms), and the 2-functor has values in the 2-category of groupoids.

More precisely, we define the lax deformation functor Def in the following way: let U = Spec(B) ∈ ob (Aff/C) be some affine scheme over C. The lax functor Def : (Aff/C) → Gpd associates to U the groupoid Def(U ) having as class of objects the B-Lie algebras and as morphisms only isomorphisms of B-Lie algebras. We forget for the moment that the Lie algebra should be isomorphic to a given one at a specified point. For a morphism f : U ′ → U in (Aff/C), we denote by f : B → B ′ the corresponding morphism of C-algebras. To f , the lax functor Def associates a functor f * : Def(U ) → Def(U ′ ) given by f * (g) = g ⊗ B B ′ (where the bracket on the tensor product is the current algebra bracket) for any B-Lie algebra g. Here B ′ is seen as a B-algebra via f .

Let us verify the axioms of a lax functor (2-functor): let g : U ′′ → U ′ be another morphism in (Aff/C) with associated morphism g :

B ′ → B ′′ of C- algebras. Then we have g * • f * ≃ (f • g) * . Indeed, (f • g) * (g) = g ⊗ B B ′′ ≃ (g ⊗ B B ′ ) ⊗ B ′ B ′′ = g * (f * (g)), where B ′′ (resp. B ′ ) is seen as a B ′ , B (resp. B-) module via g, (f • g) = g • f (resp. f ).
Now consider a third morphism h : U ′′′ → U ′′ ( h : B ′′ → B ′′′ ). We have then a commutative diagram of B ′′′ -Lie algebras

h * • g * • f * (g) = ((g ⊗ B B ′ ) ⊗ B ′ B ′′ ) ⊗ B ′′ B ′′′ / / (g ⊗ B B ′′ ) ⊗ B ′′ B ′′′ (g • h) * • f * (g) = (g ⊗ B B ′ ) ⊗ B ′ B ′′′ / / g ⊗ B B ′′′
Let us show now that our lax functor is a C-stack. It is here that the chosen Grothendieck topology on Aff/C comes into play. In order to be a C-stack, the lax functor should be on the one hand a sheaf of spaces, and on the other, one should be able to define in a unique way morphisms and objects from data given on a covering family (i.e. equivalence of descent categories). More precisely, the first condition is that for all U ∈ ob(Aff/C), and all g, h ∈ ob(Def(U )), the presheaf Hom(g, h) : Aff/C → Sets,

given by (f :

U ′ → U ) → Hom Def(U ′ ) (f * g, f * h),
should be a sheaf, i.e. one should have an exact sequence of sets

Hom Def(U) (g, h) / / Hom Def(U ′ ) (f * g, f * h) / / / / Hom Def(U ′′ ) (q * g, q * h),
which lies over the "exact" sequence of objects in Aff/C

U ′′ = U ′ × U U ′ / / / / U ′ / / U,
or, equivalently, over the "exact" sequence of C-algebras

B / / B ′ / / / / B ′′ = B ′ ⊗ B B ′ ,
q being associated to the arrow q : B → B ′′ . Let us emphasize that f : U ′ → U is here a covering family in the given Grothendieck topology, which means that the morphism f has some geometric properties, but should be moreover surjective (because one thinks U ′ as i U ′ i , and the U ′ i should cover U ). The second condition is that any descent data is effective, i.e. that for any covering family f : U ′ → U , the category Def(U ) is equivalent to the category of descent data in Def(U ′ ), i.e. to the category whose objects are the paires (x ′ , φ) of an object x ′ ∈ ob(Def(U ′ )) and an isomorphism φ : p * 1 x ′ → p * 2 x ′ , where p 1 , p 2 : U ′′ → U ′ are the two projections, such that φ satisfies the cocycle identity p * 13 φ = p * 23 φ • p * 12 φ, where p ij are projections p ij : U ′′′ → U ′′ onto a choice of two factors in the triple fibered product.

It is a non-trivial fact that the lax functor associating to an affine scheme U = Spec(B) the category of B-modules is a C-stack where Aff/C carries the fpqc topology. This is the content of Grothendieck's theorem of descente fidèlement plate [START_REF] Grothendieck | SGA 1: Revêtements étales et Groupe Fondamental[END_REF]. One easily deduces (cf Exposé VIII, §2 in [START_REF] Grothendieck | SGA 1: Revêtements étales et Groupe Fondamental[END_REF]) the following Theorem 2 The deformation lax functor Def is a C-stack.

In this form, Def is certainly not an algebraic stack, nor a differentiable stack in the sense of [START_REF] Metzler | Differentiable stacks[END_REF], as one needs a covering (algebraic) space. Nevertheless, fixing an infinite dimensional Lie algebra Der A (B) with X 0 = Spec(B), which is to be deformed, one can restrict Def to the C-stack of deformations of Der A (B) by considering first the full subcategory Aff X0 /C of affine schemes which have X 0 as fiber over a closed point. Let f : Aff X0 → Aff be the inclusion functor. Then one takes the direct image [START_REF] Giraud | Cohomologie non abélienne[END_REF] p. 83, [START_REF] Metzler | Differentiable stacks[END_REF] p. 16 f * Def as deformation lax functor, and it is a standard matter to prove that f * Def is still a C-stack (in the induced Grothendieck topology on Aff X0 . This stack f * Def is a more natural candidate for being an algebraic stack. It would be interesting to examine whether the miniversal deformation space [START_REF] Fialowski | Construction of Miniversal Deformations of Lie Algebras[END_REF] of a Lie algebra g := Der A (B) with finite dimensional H 2 (g, g) could serve as such a a covering space in order to show that in this situation, f * Def is an algebraic stack.

The morphism I

In this section, we will show that the map I gives a morphism of C-stacks from the moduli stack M g,n to the deformation stack Def, at least in any topology which is more coarse than the fppf topology (for example, in the étale topology, in which the moduli stack is usually considered).

Theorem 3

The map I defines a morphism of C-stacks from M g,n to Def in any topology which is more coarse than the fppf topology.

Proof . Recall that the moduli stack M g,n , regarded as a lax functor, associates to an affine scheme S over C the groupoid of proper flat families of curves π : C → S with sections σ 1 , . . . , σ n : S → C. The map I is now regarded as morphism of lax functors, i.e. for all affine schemes U ,

I(U ) : M g,n (U ) → Def(U )
is the functor associating to the family of curves π : C → U with sections σ 1 , . . . , σ n : U → C the Lie algebra Der A (B) such that Spec(B) = C \ △ with △ = n i=1 σ i (U ) and U = Spec(A). Observe that I(U ) is a functor here, because M g,n (U ) and Def(U ) are groupoids. Now in order to be a morphisms of stacks, one needs furthermore a compatibility on 2-morphisms, more precisely, for a every covering family f : U ′ → U in Aff/C, there exists in the diagram

M g,n (U ) I(U) / / f * Def(U ) f * M g,n (U ′ ) I(U ′ ) / / Def(U ′ ) a natural isomorphism α(f ) : f * • I(U ) → I(U ′ )
• f * between the two possible compositions of functors. This last statement is true in the fppf topology, because of lemma 1, the compatibility of the fppf topology with base change, and the fact that

(X × U U ′ ) \ △ ′ ∼ = (X \ △) × U U ′
which follows from the compatibility of the embedding X ֒→ P N with base change. This concludes the proof of theorem 3.

Remark 9

The morphism I is almost a monomorphism; indeed, for integral normal B we can apply Pursell-Shanks theory and use theorem 1. But for a general affine base scheme S = Spec(A), one cannot hope to have monomorphy of I. For example, a non-reduced scheme and its reduction may well have isomorphic tangent Lie algebra, cf remark on p. 313 in [START_REF] Hauser | Affine varieties and Lie algebras of vector fields[END_REF]. Theorem 1 shows however that I is a monomorphism for the stack of families of smooth curves.

Lie algebra cohomology sheaves on moduli stack

In this section, we will examine natural sheaves on the deformation stack Def arising from the cohomology of Lie algebras. The map I permits to pull them back to M g,n .

Let g be an A-Lie algebra, where A is some C-algebra. Let M be an Amodule carrying an A-linear action of g. Then denote by H * A (g; M ) the cohomology of g with values in M , where cochains are supposed to be A-linear. There is a base change formula for this kind of cohomology space: Lemma 5 Let A → B be a morphism of commutative unital C-algebras such that B is an A-module of finite presentation. Then

H * B (g ⊗ A B; M ⊗ A B) ∼ = H * A (g; M ) ⊗ A B.
Proof . Recall that the cochain complex computing H * A (g; M ) consists of spaces Hom A (Λ p A (g), M ), where Λ p A (g) means the skewsymmetric tensor product over A of p factors of g. The proof is easily deduced from three steps:

(g ⊗ A B) ⊗ B (g ⊗ A B) ∼ = g ⊗ A g ⊗ A B, Hom B (g ⊗ A g ⊗ A B, X) ∼ = Hom A (g ⊗ A g, X),
where X is a B-module, on the RHS viewed as an A-module, and

Hom A (g ⊗ A g, M ⊗ A B) ∼ = Hom A (g ⊗ A g, M ) ⊗ A B,
where we used in the last step the finite presentation of B as an A-module.

Corollary 2 Let B = C and A → C be a retraction of the unity C → A. Then

H * C (g ⊗ A C; M ⊗ A C) ∼ = H * A (g; M ) ⊗ A C,
meaning that the fiber of the cohomology sheaf is the cohomology of the fiber of the family of Lie algebras, cf lemma 2.

The above lemma means that the prescription

g → H * A (g; M )
for a fixed module M , defines a cartesian sheaf of vector spaces H * (M ) on the stack Def in the fppf topology, and a cartesian sheaf I * (H * (M )) on the stack M g,n . The fixed module M is supposed to be an A-module and a gmodule for any A-Lie algebra g. Examples are the ground field M = C ("trivial coefficients"), the Lie algebra itself M = g ("adjoint coefficients"), or algebraic constructions with these (like the symmetric algebra M = S * g etc).

What is known about these cohomology spaces ? It is known that H 1 (C) = 0 on the locus of smooth families, because the Lie algebra of vector fields on a smooth curve is simple, see [START_REF] Beilinson | Algebraic conformal field theory[END_REF] p. 21. I can also show that H 2 (M er k ; C) ∼ = H 1 (Σ k ), the singular cohomology of the curve Σ k := Σ {p 1 , . . . , p k } for any Lie algebra M er k of meromorphic vector fields on the compact connected Riemann surface Σ with possible pôles in the k fixed points p 1 , . . . , p k . The adjoint cohomology seems to be trivial. All these are algebraic or discrete cohomology computations. For continuous cohomology, i.e. cohomology with continuous cochains, the Lie algebras of meromorphic vector fields M er k are endowed with the subspace topology from the Lie algebra Hol(Σ k ) of all holomorphic vector fields on Σ k . The continuous cohomology of Hol(Σ k ) (all degrees and values in various modules) is known by work of Kawazumi, but it seems artificial to endow these algebraic objects with topology.

One deduces from these results that the principle that cohomology should only increase in isolated points is not true in this infinite dimensional setting. Indeed, Fialowski and Schlichenmaier loc. cit. have a family of genus 0 curves with 3 points which degenerates to the Witt algebra

Vect pol (S 1 ) = i∈Z Cx i+1 d dx .
It is known that H 2 (Vect pol (S 1 ); C) is of dimension one, but on the other hand, H 2 (M er 3 ; C) is of dimension two.

Deformations of Lie algebras arising from families of singular plane curves

There is a way to incorporate families having as some of their members non stable curves such as cusps, for example, into the picture. Up to now, we took into account only families of singular curves which constitute the boundary of the moduli stack. The other approach is to normalize the family first in a certain way and to map the thus obtained family of smooth curves into M g,n . This corresponds to the normalization which occurs in the procedure defined by Fialowski and Schlichenmaier loc. cit.. Consider families of irreducible plane curves with nodes and cusps as their only singularities. Arbitrary plane curves of degree d are parametrized by a Hilbert scheme Hilb(d) which can be identified with P N for N = d(d + 3)/2. Indeed, a curve is just represented by its homogeneous equation, up to scalars. Hilb(d) is a fine moduli space and posesses a universal family, cf [START_REF] Harris | Moduli of Curves[END_REF].

In the following, we are inspired by the introduction of [START_REF] Galati | Number of Moduli of irreducible families of plane curves with nodes and cusps[END_REF]. Let Σ d k,n ⊂ P N be the closure in the Zariski topology of the set of reduced and irreducible plane curves of degree d with k cusps and n nodes. Let Σ ⊂ Σ d k,n be an irreducible component of Σ d k,n . Denote by Σ 0 the open set of Σ of points s ∈ Σ such that Σ is smooth at s and such that s corresponds to a reduced and irreducible plane curve of degree d with n nodes and k cusps and no further singularities. The restriction of the universal family S 0 → Σ 0 is an equigeneric family of curves, i.e. the genus g of the normalizations of the fibers of the family is constant. This property implies that one gets a family of smooth curves by normalizing the total space, cf theorem (2.5) in [START_REF] Diaz | Ideals of singular plane curves[END_REF]. In this way, there is a regular map of schemes from S 0 to the scheme of moduli of curves, and a morphism of algebraic stacks S 0 → M g , where S 0 is seen as an algebraic stack via its functor of points and g = d -1 2 -k -n. One can introduce m marked points in order to get a morphism of algebraic stacks

N : S m 0 → M g,m .
The composition I • N : S m 0 → M g,m → Def gives then the construction of Fialowski and Schlichenmaier in a conceptual way: the map N consists of normalizing, i.e. desingularizing the singular fiber, and the map I consists of extracting the m marked points from the family of smooth curves and then taking the Lie algebra of vector fields tangent to the fiber.

By its construction, it is clear that the same framework applies to flat equigeneric families of projective curves with all fibers reduced, where parameter and total space are reduced separated schemes of finite type over C, see [START_REF] Diaz | Ideals of singular plane curves[END_REF] pp. 436-437.

Examples

This section is based on discussions with Martin Schlichenmaier.

(1) Let us first consider a rather trivial example. Let g be a non-abelian Lie algebra with Lie bracket [, ]. Define a global deformation of g over the affine line A = Spec(C[t]) by (g ⊗ C C[t], [, ] t ) with the bracket [, ] t = (t -1)[, ]. This is a global deformation of g which is non-trivial in the sense that there are two fibers which are non-isomorphic as C-Lie algebras (namely the fibers above t = 0 and t = 1; the Lie algebra above t = 1 is abelian). But admittedly, it is trivial in some sense.

(2) Let π : C → S be a smooth family of curves with marked points σ 1 , . . . , σ n : S → C. We explained in sections 2 and 3 how to associate to π : C → S a global deformation g A of Lie algebras of the Lie algebra g of regular vector fields on C 0 \ (σ 1 (0)∪. . .∪σ n (0)), with Spec(A) = S. In case g A is a not a free A-module, this deformation cannot be trivial (as a global deformation of Lie algebras).

(3) This is the example of Fialowski-Schlichenmaier loc. cit.. Elliptic curves can be parametrized in the complex projective plane P 2

C by e 1 , e 2 and e 3 such that the curve is given by the equation The nodal cubic is singular, but still stable and constitutes therefore a point of the boundary in the Deligne-Mumford compactification of the moduli space, while the cuspidal cubic is not. As the genus of the desingularization (i.e. normalization) decreases, the desingularization of E N and E C is the projective line P 1 C . Fialowski and Schlichenmaier extract from the above family of curves a family over C[t] and compute explicitly the relations of the Lie algebra of vector fields on the family of elliptic curves. It turns out that the generic fiber Lie algebra is not isomorphic to the Witt algebra, while the fiber at t = 0 is.

Y 2 Z = 4(X -e 1 Z)(X -
The point which makes this example less trivial than the preceeding is that in any neighborhood of 0, the restriction of the global deformation to this neighborhood remains non-trivial, while in the first two examples, there are neighborhoods where the deformation is trivial.

Fialowski and Schlichenmaier pointed out in loc. cit. that this is an example of an infinitesimally rigid Lie algebra (because the Witt algebra has trivial second cohomology space with adjoint coefficients) which has non-trivial global deformations.

The maps I and N permit in principle to define lots of examples of this third type. The hard part is to compute the relations of the so defined Lie algebras in order to render the examples explicit.

  Siebert shows that for L = Der C (B) with Y = Spec(B) irreducible of finite type over C, B(L) is an integral extension of B. Normality of B then forces B(L) = B. In this sense, Siebert does not construct an explicit isomorphism Y ≃ Y ′ , but he deduces the existence of an isomorphism from the translation of the geometric structures into Lie algebraic objects.

  e 2 Z)(X -e 3 Z) with e 1 + e 2 + e 3 = 0 and △ = 16(e 1 -e 2 ) 2 (e 1 -e 3 ) 2 (e 2 -e 3 ) 2 = 0. △ = 0 assures that the curve is non-singular. These equations form a family of curves over B := {(e 1 , e 2 , e 3 ) | e 1 + e 2 + e 3 = 0, e i = e j ∀i = j}. Completing the parameter space B to B := {(e 1 , e 2 , e 3 ) | e 1 + e 2 + e 3 = 0} admits singular cubics. Partial degeneration (e.g. e 1 = e 2 = e 3 ) leads to the nodal cubic E N : Y 2 Z = 4(X -eZ) 2 (X + 2eZ), while overall degeneration (e.g. e 1 = e 2 = e 3 ) leads to the cuspidal cubic E C : Y 2 Z = 4X 3 .

In this way we reduce at a point x the problem to the fiber π -1 (x), and Siebert's theory implies for π -1 (x) smooth that π -1 (x) = M φ , the space of maximal, finite-codimension subalgebras L φ of L φ := Der C (B⊗ A C), and furthermore that the algebra of regular functions Reg(π

In order to describe not only the fibers, but all points of Y , consider now the set of pairs

Observe that we first have to project the expression f (L φ )δ -θ onto the fiber via π φ in order to state that the result should vanish at L φ , seen as a point of

If B is a finitely generated A-algebra, there exists an epimorphism

and consequently a monomorphism

Denote by ∂ j (the isomorphic images in Der A (B n ) of) the generators of the free module B n . For a given f ∈ B, choose δ and θ such that the condition to be in B holds, and express δ = j g j ∂ j and θ = j h j ∂ j . Abusing slightly notation, we have

for all p ∈ π -1 (x), by definition of δ and θ. This means that f is a rational function on Y , regular on fibers π -1 (x) for x smooth. Therefore B ⊂ K(Y ). L is a B-module by setting f • δ = θ, for the θ specified through the definition of B by f and δ. Its annihilator is zero. On the other hand, L = Der A (B) is a finitely generated B-module if A is noetherian. By a well-known criterion (cf [START_REF] Eisenbud | Commutative algebra. With a view towards algebraic geometry[END_REF] p. 123, cor. 4.6), B is thus an integral extension of B.

In conclusion, if Y is normal, B = B is characterized by Lie theoretic methods, and the first part of the proposition follows.