text:
Talk:Regions of the Philippines

NCR
In the article it says November 7, 1975: Metropolitan Manila (National Capital Region) created But actually only Metropolitan Manila was created. PD 824 has no mention of National Capital Region. In fact, when Metro Manila was created in 1975, it was part of Region 4. Its status is similar to a province, with Imelda Marcos as Governor. In 1987, Metro Manila was elevated to a region under Cory Aquino, and its name was changed to National Capital Region (NCR). — Preceding unsigned comment added by 180.190.91.34 (talk) 14:38, 22 September 2015 (UTC)
 * Metropolitan Manila was actually declared as the NCR in 1978, also under Marcos. Added this bit and source in the article's history section.--RioHondo (talk) 14:58, 22 September 2015 (UTC)

Palawan - from Region IV-B to Region VI and back again
Readers of this page: please note that Palawan was moved to Region VI as of June 5, 2005. Essentially, Palawan is now in the Visayas and as such is no longer part of Region IV-B. -- Josh Lim 4 July 2005 13:53 (UTC)

Executive Order No. 429 dated May 23, 2005 transferred the province of Palawan from Region IV-B (MIMAROPA) to Region VI (Western Visayas). However, the transfer has not yet actually happened. Due to what I presume was some sort of back-lash, the transfer has been put on hold pending the submission of "implementation plan" as specified in Administrative Order No. 129 dated August 19, 2005. Given that the Department of the Interior and Local Government and the National Statistical Coordination Board currently group Palawan in Region IV-B (although they did both show it under Region VI briefly somewhere between EO 429 and AO 129 being issued), I think it is currently incorrect to show Palawan under Region VI rather than Region IV-B. There is also the issue of Region IV-B being renamed from MIMAROPA to MIMARO. As I can find no offical (i.e. government) reference to this name change and given that DILG and NSCB continue to use MIMAROPA, I feel this is also an error.

I suppose the current "legal status" is that Palawan is eventually going to be transferred to Region VI but hasn't yet officially.

Taiwai94 July 17, 2006, 18:07, UK time

Taken from the Palawan Page -- Meynardtengco  Palawan was transferred to Region VI (Western Visayas) on May 23, 2005 by virtue of Executive Order 429. However, Palaweños criticized the move, citing a lack of consultation, with most residents in Puerto Princesa City and all municipalities but one preferring to stay with Region IV-B.

Consequently, Administrative Order No. 129 was issued on August 19, 2005 to address this backlash. This Order directed the abeyance of Executive Order 429 pending the approval of an implementation plan for the orderly transfer of Palawan from MIMAROPA to Region VI.

Hence, Palawan is currently still part of MIMAROPA.

Storing overviews here. —seav 09:26 26 Jun 2003 (UTC)

What do you mean by overviews. This is the first time, I've visited this. Why did you not just paste them into individual pages. By the way, I copied your Central Luzon to complete the Regions. sorry. wayne manuel 01:25, 19 Aug 2003 (UTC)

National Capital Region (NCR)
Manila, the capital of the country lies within a special administrative region commonly known as Metro Manila and officially as the National Capital Region (NCR). It consists of 13 cities and 4 municipalities, and is the only region in the country that has no provinces.

Ilocos Region (Region I)
Ilocos Region is located in the northwest portion of Luzon island. Most of its inhabitants speak Iloko and are of Ilocano descent. The provinces in the region are: Ilocos Norte, Ilocos Sur, La Union, and Pangasinan. Some of its cities are Vigan, Laoag, Dagupan, and San Fernando, which is the regional capital.

Cagayan Valley (Region II)
Cagayan Valley is located in the northeast portion of Luzon. Running through its center is the country's longest river, Cagayan River. Its provinces are: Batanes, Cagayan, Isabela, Nueva Vizcaya, and Quirino. The regional capital is Tuguegarao City.

Central Luzon (Region III)
Central Luzon contains the largest plain of the country and produces most of the country's rice supply. Its provinces are: Aurora, Bataan, Bulacan, Nueva Ecija, Pampanga, Tarlac, and Zambales. San Fernando, in Pampanga, is the regional capital. Other major cities include Olongapo, Angeles, Malolos, and Tarlac City.

CALABARZON (Region IV-A)
CALABARZON, one of the newest regions of the country, was previously a part of Southern Tagalog (Region IV). It is one of the most populated areas of the country and the second most densely populated region after Metro Manila. The name of the region is actually an acronym that stands for its provinces, which are: Batangas, Cavite, Laguna, Quezon, and Rizal. Some of the chief cities are Lucena, Batangas City, Tagaytay, and San Pablo. The regional capital is Quezon City, which is in Metro Manila.

MIMAROPA (Region IV-B)
MIMAROPA, also one of the newest regions of the country, was previously a part of Southern Tagalog (Region IV). It contains most of the islands in the Luzon group. The name of the region is an acronym that stands for its provinces, which are: Marinduque, Mindoro Occidental, Mindoro Oriental, Palawan, and Romblon.

Bicol Region (Region V)
Bicol Region occupies the Bicol Peninsula at the southeastern end of Luzon island. The inhabitant call themselves Bicolanos and speak the Bicolano language. The region is where Mayon Volcano can be found. The provinces in Bicol are: Albay, Camarines Norte, Camarines Sur, Catanduanes, Masbate, and Sorsogon. The major cities in this region are Naga City and Legazpi City. Legazpi City is the regional capital.

External Link)

 * DILG Regional Office No. 5 —Preceding unsigned comment added by 210.213.136.98 (talk) 03:08, 27 October 2010 (UTC)

Cordillera Administrative Region (CAR)
CAR, located in the Cordillera mountains of Northern Luzon, is a special region for the indigenous tribes of these mountains. Some of the provinces are actually names of tribes living there. Its provinces are: Abra, Apayao,Benguet, Ifugao, Kalinga, and Mountain Province. Baguio City, which is also the regional capital, is a popular tourist destination especially among Filipinos. The famed Banaue Rice Terraces in Ifugao is another tourist attraction.

Western Visayas (Region VI)
Western Visayas consists of the island of Panay and the western half of Negros. People here primarily speak Ilonggo and a dialect of Cebuano called Binisaya. Its provinces are Aklan, Antique, Capiz, Guimaras, Iloilo, and Negros Occidental. The chief cities in this region are Iloilo City and Bacolod City. Iloilo City is the ragional capital.

Central Visayas (Region VII)
Central Visayas includes the islands of Cebu, Bohol, and Siquijor, all provinces, and the eastern half of Negros which is the province of Negros Oriental. This is the third most developed region after Metro Manila and CALABARZON. Cebu City is its regional capital and the most important city in this region. Other cities include Dumaguete, Mandaue and Tagbilaran.

Eastern Visayas (Region VIII)
Eastern Visayas consists of the islands of Leyte and Samar Most people in this region speak Waray. The region is usually the first region to be hit by typhoons being in the easternmost portion of the typhoon belt. The provinces in this region are Biliran, Leyte, Southern Leyte, Eastern Samar, Northern Samar, and Samar. Tacloban City is the regional capital.

Capitalization
Why are the names of some regions in all capitals and others in mixed case? Gdr 6 July 2005 23:52 (UTC)

 * The ones in all caps are acronyms, containing the names of their provinces or cities. TheCoffee 9 July 2005 07:11 (UTC)

I think this article needs a brief note explaining this. Otherwise it seems rather odd. Gdr 12:44:38, 2005-08-03 (UTC)

 * Okie, fixed. Coffee 13:33, 3 August 2005 (UTC)

Region Name "Nomenclature"
Throughout Wikipedia, as a general rule (NCR, CAR and ARMM are the exceptions) regions are listed with region name followed by the region number as a numeral, enclosed in brackets such as Central Visayas (Region VII). However, many websites and documents, especially from Philippine government department/bureau/agency websites list regions by the region number as a numeral then by name enclosed in brackets i.e. Region VII (Central Visayas).

It makes sense to put more emphasis on the region number than the name as the region name is subject to change whereas the number isn't such as the renaming of Region XI (Southern Mindanao) to Region XI (Davao Region). Same applies to Regions IX and XII.

Although I can't find any exact specification as to how regions should be presented, an extract from certain technical notes from PSGC codes from NSCB says:

''Regions are given a Region number and at the same time a general descriptive clause to describe the area. For example: Region 6 has a descriptive clause, Western Visayas attached to it. In such case, where the descriptive clause is changed, without any change in the regional boundary lines, the Region Code remains the same.''

I think this article really only applies to PSGC but the "general consensus" seems to indicate more emphasis on region numbers with some variations, such as Region VII, Region 7, Region VII (Central Visayas) and Region VII - Central Visayas''. Likewise, many of Ferdinand Marcos' Presidential Decrees refer to regions just by number (Region 7, Region VII). Since Marcos was the one who created regions, I think this puts argument in favour of region numbers as opposed to names. Also, Republic Act No. 7901: An Act Creating Region XIII To Be Known As The Caraga Administrative Region, And For Other Purposes.

Assuming I've made the point with region number over names, then the listings of regions on Wikipedia is therefore technically incorrect. Of course, there are the 3 odd exceptions (NCR, CAR and ARMM) which are listed by full name and then by abbreviation, such as National Capital Region (NCR) If anyone agrees with me point, shouldn't we change the way regions are listed on Wikipedia?

Taiwai94 20:23, 6 November 2005 (UTC);

 * I'd agree that government institutions would refer to the regions by number. However, I think it is more encyclopedic to use the descriptive name as the article title. I bet more people know "Bicol Region" than "Region V." As for renaming problems (e.g., Western Mindanao to Zamboanga Region), that's what the Move tool is for. For common usage, I think periodicals are a better gauge than administrative/government sites. See these Google results:

"Western   "Region 6" OR    "Ilocos     "Region 1"     CARAGA    "Region 13" OR                   Visayas"    "Region VI"      Region"   OR "Region I"             "Region XIII" Daily Inquirer     279            77           239          155           160           15 Manila Times       239            27           351          145           154            3 Manila BUlletin    207            62           121          165           427           17 Philippine Star     20             4            24           36            45            0

 * While it's not a landslide, you can see that newspapers generally prefer to refer to the region by their descriptive names rather than their designated number. Of course, you're free to do your own experimentation.

 * I believe that the status quo is okay as it is. --seav 14:27, 7 November 2005 (UTC)

Palawan
In what region is Palawan included? Is it on Region IV-B or VI? i think it should be in IV-B because it belong to MAMIROPA. —Preceding unsigned comment added by 121.54.96.2 (talk) 03:54, 29 December 2007 (UTC)

neds1OVERVIEWMAP
+thumbs2smal-----Please note, I have [[Repetitive Strain Injury]] and find typing very hard. I use a form of shorthand, which may be difficult to understand. I can be contacted through MSN (sven70) or Skype (sven0921) if my meaning is unclear. (talk) 04:47, 15 July 2010 (UTC)

Autonomous Regions
I am suggesting to make correction in the information on autonomous region.

Following Regions have its own autonomy
 * Autonomous Region for Muslim Mindanao (ARMN)
 * Caraga
 * Cordillera Administrative Region

Help me to get reference. I found in in a Elementary Book ( Grade 4) I wil look again the ISBN Bonvallite (talk) 10:33, 3 September 2011 (UTC)

In Zamboanga Peninsula, these are the following 1. Zambaonga del Norte Province 2. Zamboanga del Sur Province 3. Zamboanga Sibugay Province 4. Independent, Chartered & Highly-Urbanized City of Zamboanga 5. City Isabela

Zamboanga City was never been or is never been part of any of the 3 Provinces as mentioned.

meaning... el mapa pa lang del Filipinas es mali ya! — Preceding unsigned comment added by 83.136.59.145 (talk) 15:18, 14 January 2012 (UTC)

Map
http://wikitravel.org/upload/shared//3/3f/Map_of_Philippines.png

Rajmaan (talk) 19:56, 17 April 2014 (UTC)

Styling
This edit caught my eye. It looks to me as if some attention ought to be paid to the Manual of Style/Text formatting guideline, regarding boldface, italics and emphasis. Wtmitchell (talk) (earlier Boracay Bill) 21:53, 21 December 2015 (UTC)

MIMAROPA as Southwestern Tagalog Region
RA 10879 (lapsed into law on July 17, 2016) calls the region now as either Southwestern Tagalog Region or MIMAROPA Region (no more numerics, such as "IV-B"). Even the Philippine Statistics Authority removed the "IV-B" designation and news articles say the former "Region IV-B" has been replaced with "MIMAROPA Region" already. Xeltran (talk) 05:52, 4 January 2017 (UTC)

 * Thanks Xeltran for updating this info! I added that no boundary changes were involved and that it was in effect merely a renaming and discontinuation of the "Region IV-B" designation. Otherwise it appears as if this is a new region.
 * Based on the Act, it seems to me that the official designation is in all-caps. Should Mimaropa be moved back to MIMAROPA and all references to it changed back to all-caps? -- P 1 9 9 ✉ 14:47, 4 January 2017 (UTC)
 * Indeed it is officially written in all-caps, same goes for CALABARZON and SOCCSKSARGEN, though I think there is some sort of MOS rule that was applied that's why these are styled in title case in Wikipedia. Xeltran (talk) 03:22, 5 January 2017 (UTC)
 * & : Thanks for the update! True, they have been moved following WP:OFFICIAL and WP:COMMONNAME, just like Metro Manila (National Capital Region), Caraga (CARAGA Region/Region XIII) and Negros (Negros Island Region/Region XVIII). And following the same WP rule, i believe we should keep Region IV-B as its alternative title alongside Southwestern Tagalog Region while leaving the article at its most common name of Mimaropa. IMO, Region IV-B may not be official anymore, but it still is a common designation for Mimaropa. For instance, cars in that region still carry "Region 4B" in their plates. Cheers!--RioHondo (talk) 04:27, 5 January 2017 (UTC)

I'd just like to query references made to Southwestern Tagalog Region which appears in several places on pages and even appears as Southwestern Tagalog Region (Mimaropa). I think there's been a misunderstanding with the name. Republic Act No. 10879 clearly states:

'''Section 3. The Southwestern Tagalog Region is hereby established to be known as the MIMAROPA Region which shall be composed of. . .'''

The words to be known as constitute an explicit reference to name. If the official law states the region is to called MIMAROPA Region then I see no reason as to why Southwestern Tagalog Region is being included as part of the name. In the context of this Republic Act, I think Southwestern Tagalog Region is simply a geographic reference to the region and nothing more. That said, I think Southwestern Tagalog Region should be completely removed as part of the name except where RA 10879 is being quoted.

Take a look at the Philippine Statistical Agency.

DS94 (talk) 15:02, 21 January 2017 (UTC)

Wrong edit
I think someone made a wrong edit recently. This person added numerous component cities which should not be there in the table. For example, Malolos City is added in Central Luzon, but this city is a component city, not a highly centralized or independently governed one. I think it could be a mistake. Trung geo (talk) 16:01, 16 March 2017 (UTC)

Languages in the regions
I wasn't sure whether to raise this issue here or in Wikipedia talk:Tambayan Philippines. As it is specific to articles about regions, I decided to raise it here.

I happened to be looking at the Autonomous Region in Muslim Mindanao article, and noticed that the parameter of the infobox is set to:  Banguingui | Maguindanao | Maranao | Tagalog | Tausug | Yakan | Sama | others. It struck me that no mention is made there of the country's two official languages, Filipino and English, and that Tagalog is listed rather than Filipino (the constitutionally designated national language).

I then took a look at the Mimaropa article, and saw similar issues.

Perhaps Filipino, the national language, should be mentioned in the regional articles as a language used in the region. Also, as Filipino and English are official languages countrywide, perhaps both should be mentioned in the regional articles. Perhaps those mentions should be followed by a list of regional languages which are specific to that region. Perhaps not.

In any case, If Tagalog is mentioned in regional articles as a regional language, it probably ought to be differentiated there from Filipino. Also, the constitution designates Spanish and Arabic as auxiliary official languages countrywide for promotion on a voluntary and optional basis, and whether (and, if listed, how) those should be presented ought also to follow the same standard in all regional articles.

I think that those issues ought to be addressed in the regional articles, but perhaps it is not appropriate to duplicate info on all of those nitpicky details separately in each regional article. Perhaps a Regional languages section should be added in this article presenting this information and a see also note added to the to the infobox parameter in those articles directing attention to that.

The regional articles should all use the same standards for content and presentation. Perhaps a template (e.g., Philippine Regional Languages) might be created which would take a region name as an input parameter and would output content to be placed in the  infobox parameter of a regional articles; grouping this info for all regions together in a subtopic-specific template would simplify assuring that presentation remains consistent across all regions. such a template might also be transcluded as appropriate by other templates in Category:Languages of the Philippines templates.

I will put a comment in the talk pages of articles for all the regions listed in this article and on the Tambayan Philippines talk page, soliciting discussion here. It might take me some time to do that, as I am currently in Romblon, Romblon, and the internet is nearly unusable here due to absolutely miserable unreliability and, when usable, it is very slow. Wtmitchell (talk) (earlier Boracay Bill) 02:45, 28 March 2017 (UTC)

 * I may be wrong, but several years ago, regions' articles were using a "locally-used" infobox, but was TFDed in favor of using Infobox settlement. Presumably, we can use that infobox as the skeleton for a new infobox, along with the settlements in the Philippines with built in templates for uniformity.
 * Re: Languages -- AFAIK, there are no "official" regional languages, only "official languages" in the Philippines. Presumably "Languages" can include languages that extensively used in the region. English is almost never used on the street, unless you're at work/school/talking to a foreigner. Filipino is trickier since there's a dispute on where Tagalog ends and Filipino begins. – H T  D  09:33, 28 March 2017 (UTC)
 * Yep, no official regional languages but there are 19 recognized regional languages which is a different story. :) mcLovin'tosh (talk) 15:04, 28 March 2017 (UTC)

Frankly, languages is a mess all across Philippine geographic articles. In numerous municipality and city articles, there is a constant back and forth as to what languages to include, a constant adjusting of spoken language percentages. And all of it unreferenced/unsourced. It may be better to bring this to Wikipedia talk:Tambayan Philippines to decide what and how to include languages, and then make it part of MOS:PHIL. -- P 1 9 9 ✉ 18:35, 28 March 2017 (UTC)
 * Can't we just use the census? AFAIK the census had language data for provinces. Dunno for lower level subdivisions... – H T  D  14:24, 29 March 2017 (UTC)
 * Of course. Census info is authoritative and reliable. It should be the only source used for this. But I haven't seen this for LGU's... -- P 1 9 9 ✉ 15:02, 29 March 2017 (UTC)
 * I think only the 2000 census had info on languages at the province level. – H T  D  15:54, 29 March 2017 (UTC)
 * Lacking anything better, then that is what we'll have to use... -- P 1 9 9 ✉ 20:11, 29 March 2017 (UTC)

Boholano is not a language, it's a dialect of Cebuano. I doubt whether Porohanon is a true language either, and the whole of Camotes islands are only about 100,000 population altogether. Bantayanon is mostly Cebuano, but with underlying strata of Hiligaynon and Waray. – Not Samuel Pepys (talk) 09:18, 12 November 2020 (UTC)

This is confusing
The number after caraga or "region XIII" is very confusing "XIII" means 13 but its labeled "16", why is that? 120.28.219.20 (talk) 04:07, 27 February 2022 (UTC)

The Philippine Standard Geographic Code numbers it differently. Howard the Duck (talk) 04:11, 27 February 2022 (UTC)

title: Talk:Regions of the Philippines


text:
The Scar (novel)

The Scar is a weird fantasy novel by British writer China Miéville, the second set in his  Bas-Lag universe. The Scar won the 2003 British Fantasy Award and was shortlisted for the 2003 Arthur C. Clarke Award. The Scar was additionally nominated for the Philip K. Dick Award in 2002 and the Hugo Award for Best Novel in 2003. It is set directly after the events described in Perdido Street Station.

Plot summary
The Scar opens with the journey of a small ship which has set out from the city New Crobuzon (the setting of Perdido Street Station). It is heading to the city's new colony, Nova Esperium, which lies across the Swollen Ocean of Bas-Lag. On board the ship are:
 * Bellis Coldwine, a cold, reserved linguist who is fleeing for her life for her alleged connection to the events in Perdido Street Station.
 * Johannes Tearfly, a scientist whose interests lie in megafauna and underwater sealife.
 * Tanner Sack, a Remade criminal (that is, he has had his body surgically and magically altered as punishment for his crime) who is bound for slavery.
 * Shekel, a young cabin boy who befriends Tanner.

Before the ship reaches Nova Esperium, it is captured by pirates, and the passengers, crew and prisoners are all press-ganged into being citizens of Armada, a floating city made of thousands of ships. Tanner uses his newfound freedom to embrace his remaking. He has his body further remade and the earlier, rough work perfected, becoming an amphibious sea-creature. Treated now as an equal citizen rather than a prisoner or slave, Tanner's loyalties fiercely lie in Armada.

Bellis meanwhile despises her new life as a librarian for the city's vast collection of stolen books, and yearns for home. She gains the attention of the powerful Uther Doul, who is a bodyguard to the mysterious, scarred leaders of Armada known as the Lovers. Doul, for his own reasons, involves Bellis much more closely in the city's matters. She soon becomes privy to a plan formulated by the Lovers to raise a mythical sea creature known as the avanc. Simultaneously, she becomes involved with a New Crobuzonian spy named Silas Fennec, who reveals that the grindylow of the Cold Claw Sea are planning war on New Crobuzon. Silas was on his way home to warn his leaders of this war (thus saving the millions of innocents who might be slaughtered by the grindylow) when he was captured by Armada. Bellis and Silas find physical release in each other, and commiserate that they are powerless to save their home city.

Shekel, who is learning to read with Bellis's help, finds a strange book in the Armada's library. He brings it to Bellis, and she realizes that it is the book the Lovers need to raise the avanc. Knowing that she must get a message home, Bellis destroys a large part of the Appendix which shows the equations and methods to raise the avanc.

With the book destroyed, the Lovers seek the book's writer, Krüach Aum, the only person who knows how to summon the mythical creature. Armada mounts an expedition to his unnamed island home, which is the island of the dreaded Anophelii (a horrific and deadly race of mosquito-people). The Lovers find Aum and the information they need, while Bellis uses their time on the island, away from Armada, to get a message home warning of the impending grindylow invasion. Armada then successfully raises the avanc and captures it – no mean feat, as the avanc is an immense creature, several miles long. The Lovers' true plan is finally revealed: to use the great speed and pulling power of the avanc to find the fabled Scar, a place in the world where reality breaks down and anything is possible. The Lovers see this as a source of ultimate power.

On the journey far into unmapped waters, numerous matters threaten the city. Silas Fennec's actions, which have been far from honest, single-handedly bring down the fury of the New Crobuzon navy and the inhuman wrath of the grindylows. Following this, a civil war breaks out within the city. Terribly damaged, Armada finally nears the Scar, and faces the unsettling horrors that accompany the breakdown of possibility. They pick up a shipwrecked friend from a different train of possibilities, chances and choices, who warns them of The Scar and that he saw the city fall into the wound of the world and everybody was killed. Mutiny follows and the people of Armada finally force the city to turn around and head back to the Swollen Ocean, the life of quiet and piracy, that they all want.

The last chapter of the book is another excerpt of Bellis's letter home, in which she realizes how much she was being used by Doul. The reader is left to question how much Uther Doul actually was in control of the events in the book, which events were simply chance and which were 'planned' possibility.

Reception
Steven Poole reviewed the book for The Guardian and struggled with the writing style at the beginning of the book, "All through the first 40 pages or so you can hear the grunts of a writer straining too hard for effect", but "[o]nce the novel settles down after its ill-judged beginning, Miéville begins to construct an intriguing plot of espionage and deceit". He concludes that "The Scar eventually demonstrates enough invention and brutal energy, firmly ruled by a calm architectonic intelligence, to show that Miéville is one of the most imaginative young writers around in any kind of fiction."

Awards

 * British Science Fiction Award nominee, 2002
 * Philip K. Dick Award nominee, 2002
 * British Fantasy Award winner, 2003
 * Locus Award winner, 2003
 * Arthur C. Clarke Award nominee, 2003
 * Hugo Award nominee, 2003
 * World Fantasy Award nominee, 2003

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: The Scar (novel)


text:
Confusion (disambiguation)

Confusion is the state of being bewildered or unclear in one's mind about something.

Confusion or Confused may also refer to:

Geography

 * Confusion Bay, Newfoundland and Labrador, a bay in Canada
 * Confusion Island, a subantarctic island
 * Confusion Lake, a lake in Idaho, U.S.
 * Confusion Range, a mountain range in Utah, U.S.
 * Confusion Hill, a roadside attraction in Piercy, California, U.S.

Books

 * The Confusion, a 2004 novel by Neal Stephenson
 * Confusion (novella), a 1927 novella by Stefan Zweig
 * Confusions, a 1974 play by Alan Ayckbourn

Music

 * Confusion (album), an album by Fela Kuti

Songs

 * "Confusion" (Electric Light Orchestra song)
 * "Confusion" (New Order song)
 * "Confusion" (The Zutons song)
 * "Confused", by SvenDeeKay
 * "Confused" (song), by Tevin Campbell
 * "Confusion", by Alice in Chains from Facelift
 * "Confusion", by Camouflage from Relocated
 * "Confusion", by Metallica from Hardwired... to Self-Destruct
 * "Confusion", by Miz from Say It's Forever
 * "Confusion", by Sparks from Big Beat
 * "Confusion", by Zee from Identity
 * "Confused", by Angel Witch from Angel Witch
 * "Confused!", by Kid Cudi from Speedin' Bullet 2 Heaven

Other

 * ConFusion, an annual science fiction convention in Detroit, Michigan, U.S.
 * Confused.com, a British insurance and financial services comparison service
 * Apamea furva or confused, a species of moth
 * Confusion, a season of the Discordian calendar
 * "Confused" (Cow and Chicken), a television episode
 * "Confusion", an episode of Code Lyoko: Evolution
 * Confusion and diffusion (in cryptography)

title: Confusion (disambiguation)


text:
Talk:Economy of Finland

Major POV
This article appears extremely POV, in that it ignores the very significant role of the state (e.g., regulation and public expenditures) in building Finland's strong economy. It ignores the fact that Finland's highly-educated population and low-crime rate were and are supported by very high taxes and rates of expenditure (as compared.,e.g., with the U.S.) If you were to rely on this article alone, you would think that Finland's economy grew as a result of the government's simply getting out of the way - nonsense. — Preceding unsigned comment added by 204.40.1.129 (talk) 22:07, 5 January 2010 (UTC)
 * The US are an anomality. Looking at List of countries by tax revenue as percentage of GDP the Finnish taxation is in line with the taxation in other European countries. Lumialover2 (talk) 08:37, 30 August 2012 (UTC)

The homicide rate of Finland is very high compared to the other western Europe. So is it with other crimes. http://en.wikipedia.org/wiki/List_of_countries_by_intentional_homicide_rate

Finland's economy is not strong. The GDP growth of Finland was -8.5 % in 2009, -0.8 % in 2012 and about -1.0 % in 2013. Public debt of Finland is increasing very strongly. http://www.stat.fi/til/vtp/2012/vtp_2012_2013-07-11_tau_001_en.html http://www.taloussanomat.fi/porssi/talouden-tunnusluvut/ — Preceding unsigned comment added by 85.157.177.25 (talk) 08:28, 27 January 2014 (UTC)

GDP Update
Updating some GDP growth figures (according to the IMF ) and inserting a table to show growth:

http://www.stat.fi/til/vtp/2012/vtp_2012_2013-07-11_tau_001_en.html Link for verification --JDnCoke 18:08, 22 September 2005 (UTC)

Comments
The article says: "Unemployment continues to be a problem for Finland even if rapid growth of the production after the recession and active employment policies have reduced unemployment to 8,8 % of labor force in 2004. According to Bank of Finland's statistics, the estimated total rate of unemployment including hidden unemployment accounted to 18% in the fall of 2003."

Perhaps it could be noted that 8,8% - not 18% - is the official figure measured following the accepted EU standards.

"Timber and metalworking are Finland's main industries"

What about telecommunications and electronics?

Taking into account that agriculture is about 4% of the GDP it clearly receives a bit too much attention.

"Although free education through the university level also is available, only about one child in four receives a higher education in the highly competitive system." and on the other hand: ". About 60% select the academic line." Is there a discrepancy?

"The good infrastructure in Finland has enhanced Finland's position as a gateway to Russia."

The road to St. Petersburg hardly represents good infrasturcture. — Preceding unsigned comment added by 80.74.196.17 (talk) 19:04, 24 February 2006 (UTC)

Anyway, Finland is now the most ICT specialized (or knowledge based) economy in the world. That's certainly one of the most important things about Finland's economy.

http://info.worldbank.org/etools/docs/library/201645/Finland_ES.pdf — Preceding unsigned comment added by Tomi Ahti (talk • contribs) 18:28, 26 February 2006 (UTC)

There is no phone mill in Finland any more. Finland exports almost no phones any more. Microsoft acquire Nokia phones in 2013. http://www.engadget.com/2013/12/04/microsofts-nokia-acquisition-now-a-done-deal-with-final-eu-appr/.

The export of electronics has strongly diminished during last couples of years.

Metalworking in Finland is also badly diminishing because of heavy outsourcing to Eastern Europe specially to Poland and Estonia. The industry has lost 90,000 workplaces in the last ten years. — Preceding unsigned comment added by 85.157.177.25 (talk) 08:43, 27 January 2014 (UTC)

What is university education?

 * Although free education through the university level also is available, only about one child in four receives a higher education in the highly competitive system.

Isn't it so in the Anglo-Saxon countries that "career colleges", that is, the same as Finnish second-level vocational schools, are considered "universities"? The statement above is highly misleading. Currently, there are too many academically educated people, but too few vocationally educated, compared to market demand. --Vuo 15:26, 28 May 2006 (UTC)

Better article in Finnish
There is a better article on the Finnish Wikipedia. I would suggest a complete translation from that article rather than editing this version. The English article suffers from several, difficult-to-correct problems with emphasis, style, and accuracy. My first impression is that it's very similar to the CIA World Factbook entry. Missing from the article are things like: All of this should be in both Finnish and English articles. --Vuo 16:00, 19 July 2006 (UTC)
 * Finland is one of the most knowledge-based economies in the world, it's not just manufacturing, but technology.
 * The chemical industry is the third largest industry
 * Taxation facts and government spending
 * Unemployment

"Largely Free Market"?
Looking at the statistics from OECD, I am not sure how well that qualifies? http://stats.oecd.org/wbos/viewhtml.aspx?queryname=313&querytype=view&lang=en -- Acksyning 16:46, 23 August 2007 (UTC)

The looney faction could keep their hands off
AKA the market fundamentalists. And not only here. Socialist fundamentalist could also keep their hands off here. —Preceding unsigned comment added by 78.27.67.160 (talk) 19:08, 25 July 2008 (UTC)

Bot report : Found duplicate references !
In the last revision I edited, I found duplicate named references, i.e. references sharing the same name, but not having the same content. Please check them, as I am not able to fix them automatically :) DumZiBoT (talk) 16:51, 1 August 2008 (UTC)
 * "oecd2004" :
 * Finland Economy 2004, OECD
 * Economic Survey of Finland in 2004, OECD

Vandalism - can't figure out how to fix it
The sidebar on the rights contains the sentence "Micah's a fag!!!!!!!!!!!". I couldn't figure out how to delete it. Hopefully somebody more skilled in the ways of Wikipedia will see this and remove it.

83.145.240.248 (talk) 10:17, 24 September 2008 (UTC)

Finnish industry
The son of a friend of mine was doing a geography project on Finnish industry and commented this was not mentioned on this page so I have added his researches and references to this page on his behalf. I am no expert on the subject so maybe someone else will take up the baton and run Mjchesnel (talk) 10:51, 8 December 2008 (UTC)

i updated the imports and exports
https://www.cia.gov/library/publications/the-world-factbook/geos/fi.html i used the cia data--Noelmantra (talk) 16:00, 3 December 2012 (UTC)

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External link to tariff data
Hello everyone, I am working for the International Trade Centre (ITC), a UN/WTO agency that aims to promote sustainable economic development through trade promotion. I would like to propose the addition of an external link (http://www.macmap.org/QuickSearch/FindTariff/FindTariff.aspx?subsite=open_access&country=246&source=1|ITC) that leads directly to our online database of customs tariffs applied by Finland. Visitors can easily look up market access information for Finland by selecting the product and partner of their interest. I would like you to consider this link under the WP:ELYES #3 prescriptions. Moreover, the reliability and the pertinence of this link can be supported by the following facts 1) ITC is part of the United Nations, and aims to share trade and market access data on by country and product as a global public good 2) No registration is required to access this information 3) Market access data (Tariffs and non-tariff measures) are regularly updated

Thank you, Divoc (talk) 09:06, 28 December 2015 (UTC)

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title: Talk:Economy of Finland


text:
Talk:Financial mathematics

is portfolio optimization etc. considered part of mathematical finance?
 * Yes it is, IMO. Not tight at the heart of things perhaps, but certainly more so than some of the articles we have listed on this page! Pete 07:59 27 Jun 2003 (UTC)

title: Talk:Financial mathematics


text:
Quantum Quality Productions

Quantum Quality Productions (also known by their initials QQP) was a computer games company specializing in strategy games and war games.

Run by Bruce Williams Zaccagnino and Mark Baldwin it produced a number of games that achieved "cult status", most prominently The Perfect General.

Computer Gaming World reported in March 1994 that QQP was as "a very satisfying wellspring of entertainment", although its "fine games" had "below average documentation". In 1994, due to financial difficulties, QQP accepted a buy-out from American Laser Games. ALG "unceremoniously" closed the studio in December 1995, according to Computer Game Review.

Partial list of games produced by QQP

 * Battles in Time
 * Battles of Destiny
 * Bridge Olympiad
 * Conquered Kingdoms
 * Dealer's Choice Collection
 * Erben des Throns
 * Grandest Fleet
 * Heirs to the Throne (German import)
 * Lucky's Casino Adventure
 * Lost Admiral
 * Merchant Prince
 * Perfect General II
 * Perfect General
 * Pure Wargame
 * Solitaire's Journey
 * The Red Crystal: The Seven Secrets of Life
 * WWII: Battles of the South Pacific
 * Zig Zag

title: Quantum Quality Productions


text:
Dimension of an algebraic variety

In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways.

Some of these definitions are of geometric nature, while some other are purely algebraic and rely on commutative algebra. Some are restricted to algebraic varieties while others apply also to any algebraic set. Some are intrinsic, as independent of any embedding of the variety into an affine or projective space, while other are related to such an embedding.

Dimension of an affine algebraic set
Let $K$ be a field, and $L ⊇ K$ be an algebraically closed extension.

An affine algebraic set $V$ is the set of the common zeros in $L^{n}$ of the elements of an ideal $I$ in a polynomial ring $$R=K[x_1, \ldots, x_n].$$ Let $$A=R/I$$ be the K-algebra of the polynomial functions over $V$. The dimension of $V$ is any of the following integers. It does not change if $K$ is enlarged, if $L$ is replaced by another algebraically closed extension of $K$ and if $I$ is replaced by another ideal having the same zeros (that is having the same radical). The dimension is also independent of the choice of coordinates; in other words it does not change if the $x_{i}$ are replaced by linearly independent linear combinations of them.

The dimension of $V$ is This definition generalizes a property of the dimension of a Euclidean space or a vector space. It is thus probably the definition that gives the easiest intuitive description of the notion. This is the transcription of the preceding definition in the language of commutative algebra, the Krull dimension being the maximal length of the chains $$p_0\subset p_1\subset \ldots \subset p_d$$ of prime ideals of $V$.
 * The maximal length $$d$$ of the chains $$V_0\subset V_1\subset \ldots \subset V_d$$ of distinct nonempty (irreducible) subvarieties of $A$.
 * The Krull dimension of the coordinate ring $A$.
 * The maximal Krull dimension of the local rings at the points of $V$.

This definition shows that the dimension is a local property if $$V$$is irreducible. If $$V$$is irreducible, it turns out that all the local rings at points of $V$ have the same Krull dimension (see ); thus:

 * If $V$ is a variety, the Krull dimension of the local ring at any point of $V$

This rephrases the previous definition into a more geometric language. This relates the dimension of a variety to that of a differentiable manifold. More precisely, if $V$ if defined over the reals, then the set of its real regular points, if it is not empty, is a differentiable manifold that has the same dimension as a variety and as a manifold. This is the algebraic analogue to the fact that a connected manifold has a constant dimension. This can also be deduced from the result stated below the third definition, and the fact that the dimension of the tangent space is equal to the Krull dimension at any non-singular point (see Zariski tangent space). This definition is not intrinsic as it apply only to algebraic sets that are explicitly embedded in an affine or projective space. This the algebraic translation of the preceding definition. This is the algebraic translation of the fact that the intersection of $n – d$ general hypersurfaces is an algebraic set of dimension $d$. This allows, through a Gröbner basis computation to compute the dimension of the algebraic set defined by a given system of polynomial equations. Moreover, the dimension is not changed if the polynomials of the Gröbner basis are replaced with their leading monomials, and if these leading monomials are replaced with their radical (monomials obtained by removing exponents). So: This allows to prove easily that the dimension is invariant under birational equivalence.
 * The maximal dimension of the tangent vector spaces at the non singular points of $V$.
 * If $V$ is a variety, the dimension of the tangent vector space at any non singular point of $V$.
 * The number of hyperplanes or hypersurfaces in general position which are needed to have an intersection with $V$ which is reduced to a nonzero finite number of points.
 * The maximal length of a regular sequence in the coordinate ring $A$.
 * The difference between $n$ and the maximal length of the regular sequences contained in $I$.
 * The degree of the Hilbert polynomial of $A$.
 * The degree of the denominator of the Hilbert series of $A$.
 * The Krull dimension of the Stanley–Reisner ring $$R/J$$ where $$J$$ is the radical of the initial ideal of $$I$$ for any admissible monomial ordering (the initial ideal of $$I$$ is the set of all leading monomials of elements of $$I$$).
 * ''The dimension of the simplicial complex defined by this Stanley–Reisner ring.
 * If $I$ is a prime ideal (i.e. $V$ is an algebraic variety), the transcendence degree over $K$ of the field of fractions of $A$.

Dimension of a projective algebraic set
Let $V$ be a projective algebraic set defined as the set of the common zeros of a homogeneous ideal $I$ in a polynomial ring $$R=K[x_0, x_1, \ldots, x_n]$$ over a field $K$, and let $A=R/I$ be the graded algebra of the polynomials over V.

All the definitions of the previous section apply, with the change that, when $A$ or $I$ appear explicitly in the definition, the value of the dimension must be reduced by one. For example, the dimension of $V$ is one less than the Krull dimension of $A$.

Computation of the dimension
Given a system of polynomial equations over an algebraically closed field $$K$$, it may be difficult to compute the dimension of the algebraic set that it defines.

Without further information on the system, there is only one practical method, which consists of computing a Gröbner basis and deducing the degree of the denominator of the Hilbert series of the ideal generated by the equations.

The second step, which is usually the fastest, may be accelerated in the following way: Firstly, the Gröbner basis is replaced by the list of its leading monomials (this is already done for the computation of the Hilbert series). Then each monomial like $${x_1}^{e_1}\cdots {x_n}^{e_n}$$ is replaced by the product of the variables in it: $$x_1^{\min (e_1,1)}\cdots x_n^{\min(e_n,1)}.$$ Then the dimension is the maximal size of a subset S of the variables, such that none of these products of variables depends only on the variables in S.

This algorithm is implemented in several computer algebra systems. For example in Maple, this is the function Groebner[HilbertDimension], and in Macaulay2, this is the function dim.

Real dimension
The real dimension of a set of real points, typically a semialgebraic set, is the dimension of its Zariski closure. For a semialgebraic set $S$, the real dimension is one of the following equal integers:
 * The real dimension of $$S$$ is the dimension of its Zariski closure.
 * The real dimension of $$S$$ is the maximal integer $$d$$ such that there is a homeomorphism of $$[0,1]^d$$ in $$S$$.
 * The real dimension of $$S$$ is the maximal integer $$d$$ such that there is a projection of $$S$$ over a $$d$$-dimensional subspace with a non-empty interior.

For an algebraic set defined over the reals (that is defined by polynomials with real coefficients), it may occur that the real dimension of the set of its real points is smaller than its dimension as a semi algebraic set. For example, the algebraic surface of equation $$x^2+y^2+z^2=0$$ is an algebraic variety of dimension two, which has only one real point (0, 0, 0), and thus has the real dimension zero.

The real dimension is more difficult to compute than the algebraic dimension. For the case of a real hypersurface (that is the set of real solutions of a single polynomial equation), there exists a probabilistic algorithm to compute its real dimension.

title: Dimension of an algebraic variety


text:
Pavel Cherenkov

Pavel Alekseyevich Cherenkov (Па́вел Алексе́евич Черенко́в ; July 28, 1904 – January 6, 1990) was a Soviet physicist who shared the Nobel Prize in physics in 1958 with Ilya Frank and Igor Tamm for the discovery of Cherenkov radiation, made in 1934.

Biography
Cherenkov was born in 1904 to Alexey Cherenkov and Mariya Cherenkova in the small village of Novaya Chigla. This town is in present-day Voronezh Oblast, Russia.

In 1928, he graduated from the Department of Physics and Mathematics of Voronezh State University. In 1930, he took a post as a senior researcher in the Lebedev Physical Institute. That same year he married Maria Putintseva, daughter of A.M. Putintsev, a Professor of Russian Literature. They had a son, Alexey, and a daughter, Yelena.

Cherenkov was promoted to section leader, and in 1940 was awarded the degree of Doctor of Physico-Mathematical Sciences. In 1953, he was confirmed as Professor of Experimental Physics. Starting in 1959, he headed the institute's photo-meson processes laboratory. He remained a professor for fourteen years. In 1970, he became Academician of the USSR Academy of Sciences.

Cherenkov died in Moscow on January 6, 1990, and was buried in Novodevichy Cemetery.

Discoveries in physics
In 1934, while working under S. I. Vavilov, Cherenkov observed the emission of blue light from a bottle of water subjected to radioactive bombardment. This phenomenon, associated with charged subatomic particles moving at velocities greater than the phase velocity of light, proved to be of great importance in subsequent experimental work in nuclear physics, and for the study of cosmic rays. Eponymously, it was dubbed the Cherenkov effect, as was the Cherenkov detector, which has become a standard piece of equipment in particle-physics research for observing the existence and velocity of high-speed particles. Such a device was installed in Sputnik 3.

Pavel Cherenkov also shared in the development and construction of electron accelerators and in the investigation of photo-nuclear and photo-meson reactions.

Awards and honours
Cherenkov was awarded two Stalin Prizes, the first in 1946, sharing the honor with Vavilov, Frank and Tamm, and another in 1952. He was also awarded the USSR State Prize in 1977. In 1958, he was awarded the Nobel Prize in Physics for the discovery of the Cherenkov effect. He was also awarded the Soviet Union's Hero of Socialist Labour title in 1984. Cherenkov was a member of the Communist Party of the Soviet Union.

In popular culture
The novel  Ghost Fleet makes the claim that many believe the Star Trek character Pavel Chekov is named after Pavel Cherenkov.

In Starship Troopers spaceships travel faster than light using Cherenkov Drive.

title: Pavel Cherenkov


text:
St. Albans, Vermont

St. Albans, Vermont may refer to:

 * St. Albans (town), Vermont, established 1763, a town in Franklin County, Vermont, U.S.
 * St. Albans (city), Vermont, established 1902, a city in Franklin County, Vermont, U.S.

title: St. Albans, Vermont


text:
Algebraic curve

In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation $h(x, y, t) = 0$ can be restricted to the affine algebraic plane curve of equation $h(x, y, 1) = 0$. These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered.

If the defining polynomial of a plane algebraic curve is irreducible, then one has an irreducible plane algebraic curve. Otherwise, the algebraic curve is the union of one or several irreducible curves, called its components, that are defined by the irreducible factors.

More generally, an algebraic curve is an algebraic variety of dimension one. (In some contexts, an algebraic set of dimension one is also called an algebraic curve, but this will not be the case in this article.) Equivalently, an algebraic curve is an algebraic variety that is birationally equivalent to an irreducible algebraic plane curve. If the curve is contained in an affine space or a projective space, one can take a projection for such a birational equivalence.

These birational equivalences reduce most of the study of algebraic curves to the study of algebraic plane curves. However, some properties are not kept under birational equivalence and must be studied on non-plane curves. This is, in particular, the case for the degree and smoothness. For example, there exist smooth curves of genus 0 and degree greater than two, but any plane projection of such curves has singular points (see Genus–degree formula).

A non-plane curve is often called a space curve or a skew curve.

In Euclidean geometry
An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation p(x, y) = 0. This equation is often called the implicit equation of the curve, in contrast to the curves that are the graph of a function defining explicitly y as a function of x.

With a curve given by such an implicit equation, the first problems are to determine the shape of the curve and to draw it. These problems are not as easy to solve as in the case of the graph of a function, for which y may easily be computed for various values of x. The fact that the defining equation is a polynomial implies that the curve has some structural properties that may help in solving these problems.

Every algebraic curve may be uniquely decomposed into a finite number of smooth monotone arcs (also called branches) sometimes connected by some points sometimes called "remarkable points", and possibly a finite number of isolated points called acnodes. A smooth monotone arc is the graph of a smooth function which is defined and monotone on an open interval of the x-axis. In each direction, an arc is either unbounded (usually called an infinite arc) or has an endpoint which is either a singular point (this will be defined below) or a point with a tangent parallel to one of the coordinate axes.

For example, for the Tschirnhausen cubic, there are two infinite arcs having the origin (0,0) as of endpoint. This point is the only singular point of the curve. There are also two arcs having this singular point as one endpoint and having a second endpoint with a horizontal tangent. Finally, there are two other arcs each having one of these points with horizontal tangent as the first endpoint and having the unique point with vertical tangent as the second endpoint. In contrast, the sinusoid is certainly not an algebraic curve, having an infinite number of monotone arcs.

To draw an algebraic curve, it is important to know the remarkable points and their tangents, the infinite branches and their asymptotes (if any) and the way in which the arcs connect them. It is also useful to consider the inflection points as remarkable points. When all this information is drawn on a sheet of paper, the shape of the curve usually appears rather clearly. If not, it suffices to add a few other points and their tangents to get a good description of the curve.

The methods for computing the remarkable points and their tangents are described below in the section Remarkable points of a plane curve.

Plane projective curves
It is often desirable to consider curves in the projective space. An algebraic curve in the projective plane or plane projective curve is the set of the points in a projective plane whose projective coordinates are zeros of a homogeneous polynomial in three variables P(x, y, z).

Every affine algebraic curve of equation p(x, y) = 0 may be completed into the projective curve of equation $$^hp(x,y,z)=0,$$ where $$^hp(x,y,z)=z^{\deg(p)}p\left(\frac{x}{z},\frac{y}{z}\right)$$ is the result of the homogenization of p. Conversely, if P(x, y, z) = 0 is the homogeneous equation of a projective curve, then P(x, y, 1) = 0 is the equation of an affine curve, which consists of the points of the projective curve whose third projective coordinate is not zero. These two operations are reciprocal one to the other, as $$^hp(x,y,1)=p(x,y)$$ and, if p is defined by $$p(x,y)=P(x,y,1)$$, then $$^hp(x,y,z)=P(x,y,z),$$ as soon as the homogeneous polynomial P is not divisible by z.

For example, the projective curve of equation x2 + y2 − z2 is the projective completion of the unit circle of equation x2 + y2 − 1 = 0.

This implies that an affine curve and its projective completion are the same curves, or, more precisely that the affine curve is a part of the projective curve that is large enough to well define the "complete" curve. This point of view is commonly expressed by calling "points at infinity" of the affine curve the points (in finite number) of the projective completion that do not belong to the affine part.

Projective curves are frequently studied for themselves. They are also useful for the study of affine curves. For example, if p(x, y) is the polynomial defining an affine curve, beside the partial derivatives $$ p'_x$$ and $$ p'_y$$, it is useful to consider the derivative at infinity $$ p'_\infty(x,y)={^hp'_z(x,y,1)}.$$

For example, the equation of the tangent of the affine curve of equation p(x, y) = 0 at a point (a, b) is $$xp'_x(a,b)+yp'_y(a,b)+p'_\infty(a,b)=0.$$

Remarkable points of a plane curve
In this section, we consider a plane algebraic curve defined by a bivariate polynomial p(x, y) and its projective completion, defined by the homogenization $$P(x,y,z)= {}^hp(x,y,z)$$ of p.

Intersection with a line
Knowing the points of intersection of a curve with a given line is frequently useful. The intersection with the axes of coordinates and the asymptotes are useful to draw the curve. Intersecting with a line parallel to the axes allows one to find at least a point in each branch of the curve. If an efficient root-finding algorithm is available, this allows to draw the curve by plotting the intersection point with all the lines parallel to the y-axis and passing through each pixel on the x-axis.

If the polynomial defining the curve has a degree d, any line cuts the curve in at most d points. Bézout's theorem asserts that this number is exactly d, if the points are searched in the projective plane over an algebraically closed field (for example the complex numbers), and counted with their multiplicity. The method of computation that follows proves again this theorem, in this simple case.

To compute the intersection of the curve defined by the polynomial p with the line of equation ax+by+c = 0, one solves the equation of the line for x (or for y if a = 0). Substituting the result in p, one gets a univariate equation q(y) = 0 (or q(x) = 0, if the equation of the line has been solved in y), each of whose roots is one coordinate of an intersection point. The other coordinate is deduced from the equation of the line. The multiplicity of an intersection point is the multiplicity of the corresponding root. There is an intersection point at infinity if the degree of q is lower than the degree of p; the multiplicity of such an intersection point at infinity is the difference of the degrees of p and q.

Tangent at a point
The tangent at a point (a, b) of the curve is the line of equation $$(x-a)p'_x(a,b)+(y-b)p'_y(a,b)=0$$, like for every differentiable curve defined by an implicit equation. In the case of polynomials, another formula for the tangent has a simpler constant term and is more symmetric:

$$xp'_x(a,b)+yp'_y(a,b)+p'_\infty(a,b)=0,$$

where $$p'_\infty(x,y)=P'_z(x,y,1)$$ is the derivative at infinity. The equivalence of the two equations results from Euler's homogeneous function theorem applied to P.

If $$p'_x(a,b)=p'_y(a,b)=0,$$ the tangent is not defined and the point is a singular point.

This extends immediately to the projective case: The equation of the tangent of at the point of projective coordinates (a:b:c) of the projective curve of equation P(x, y, z) = 0 is

$$xP'_x(a,b,c)+yP'_y(a,b,c)+zP'_z(a,b,c)=0,$$

and the points of the curves that are singular are the points such that

$$P'_x(a,b,c)=P'_y(a,b,c)=P'_z(a,b,c)=0.$$

(The condition P(a, b, c) = 0 is implied by these conditions, by Euler's homogeneous function theorem.)

Asymptotes
Every infinite branch of an algebraic curve corresponds to a point at infinity on the curve, that is a point of the projective completion of the curve that does not belong to its affine part. The corresponding asymptote is the tangent of the curve at that point. The general formula for a tangent to a projective curve may apply, but it is worth to make it explicit in this case.

Let $$p=p_d+\cdots+p_0$$ be the decomposition of the polynomial defining the curve into its homogeneous parts, where pi is the sum of the monomials of p of degree i. It follows that $$P={^hp}=p_d+zp_{d-1}+\cdots+z^dp_0$$ and $$P'_z(a,b,0) =p_{d-1}(a,b).$$

A point at infinity of the curve is a zero of p of the form (a, b, 0). Equivalently, (a, b) is a zero of pd. The fundamental theorem of algebra implies that, over an algebraically closed field (typically, the field of complex numbers), pd factors into a product of linear factors. Each factor defines a point at infinity on the curve: if bx − ay is such a factor, then it defines the point at infinity (a, b, 0). Over the reals, pd factors into linear and quadratic factors. The irreducible quadratic factors define non-real points at infinity, and the real points are given by the linear factors. If (a, b, 0) is a point at infinity of the curve, one says that (a, b) is an asymptotic direction. Setting q = pd the equation of the corresponding asymptote is $$xq'_x(a,b)+yq'_y(a,b)+p_{d-1}(a,b)=0.$$

If $$q'_x(a,b)=q'_y(a,b)=0$$ and $$p_{d-1}(a,b)\neq 0,$$ the asymptote is the line at infinity, and, in the real case, the curve has a branch that looks like a parabola. In this case one says that the curve has a parabolic branch. If $$q'_x(a,b)=q'_y(a,b)=p_{d-1}(a,b)=0,$$ the curve has a singular point at infinity and may have several asymptotes. They may be computed by the method of computing the tangent cone of a singular point.

Singular points
The singular points of a curve of degree d defined by a polynomial p(x,y) of degree d are the solutions of the system of equations: $$p'_x(x,y)=p'_y(x,y)=p(x,y)=0.$$ In characteristic zero, this system is equivalent to $$p'_x(x,y)=p'_y(x,y)=p'_\infty(x,y)=0,$$ where, with the notation of the preceding section, $$p'_\infty(x,y)=P'_z(x,y,1).$$ The systems are equivalent because of Euler's homogeneous function theorem. The latter system has the advantage of having its third polynomial of degree d-1 instead of d.

Similarly, for a projective curve defined by a homogeneous polynomial P(x,y,z) of degree d, the singular points have the solutions of the system $$P'_x(x,y,z)=P'_y(x,y,z)=P'_z(x,y,z)=0$$ as homogeneous coordinates. (In positive characteristic, the equation $$P(x,y,z)$$ has to be added to the system.)

This implies that the number of singular points is finite as long as p(x,y) or P(x,y,z) is square free. Bézout's theorem implies thus that the number of singular points is at most (d − 1)2, but this bound is not sharp because the system of equations is overdetermined. If reducible polynomials are allowed, the sharp bound is d(d − 1)/2, this value is reached when the polynomial factors in linear factors, that is if the curve is the union of d lines. For irreducible curves and polynomials, the number of singular points is at most (d − 1)(d − 2)/2, because of the formula expressing the genus in term of the singularities (see below). The maximum is reached by the curves of genus zero whose all singularities have multiplicity two and distinct tangents (see below).

The equation of the tangents at a singular point is given by the nonzero homogeneous part of the lowest degree in the Taylor series of the polynomial at the singular point. When one changes the coordinates to put the singular point at the origin, the equation of the tangents at the singular point is thus the nonzero homogeneous part of the lowest degree of the polynomial, and the multiplicity of the singular point is the degree of this homogeneous part.

Analytic structure
The study of the analytic structure of an algebraic curve in the neighborhood of a singular point provides accurate information of the topology of singularities. In fact, near a singular point, a real algebraic curve is the union of a finite number of branches that intersect only at the singular point and look either as a cusp or as a smooth curve.

Near a regular point, one of the coordinates of the curve may be expressed as an analytic function of the other coordinate. This is a corollary of the analytic implicit function theorem, and implies that the curve is smooth near the point. Near a singular point, the situation is more complicated and involves Puiseux series, which provide analytic parametric equations of the branches.

For describing a singularity, it is worth to translate the curve for having the singularity at the origin. This consists of a change of variable of the form $$X=x-a, Y=y-b,$$ where $$a, b$$ are the coordinates of the singular point. In the following, the singular point under consideration is always supposed to be at the origin.

The equation of an algebraic curve is $$f(x,y)=0, $$ where $f$ is a polynomial in $x$ and $y$. This polynomial may be considered as a polynomial in $y$, with coefficients in the algebraically closed field of the Puiseux series in $x$. Thus $f$ may be factored in factors of the form $$y-P(x),$$ where $P$ is a Puiseux series. These factors are all different if $f$ is an irreducible polynomial, because this implies that $f$ is square-free, a property which is independent of the field of coefficients.

The Puiseux series that occur here have the form $$P(x)=\sum_{n=n_0}^\infty a_nx^{n/d},$$ where $d$ is a positive integer, and $n_0$ is an integer that may also be supposed to be positive, because we consider only the branches of the curve that pass through the origin. Without loss of generality, we may suppose that $d$ is coprime with the greatest common divisor of the $n$ such that $a_n \ne 0$ (otherwise, one could choose a smaller common denominator for the exponents).

Let $\omega_d$ be a primitive $d$th root of unity. If the above Puiseux series occurs in the factorization of $f(x,y)=0$, then the $d$ series $$P_i(x)=\sum_{n=n_0}^\infty a_n\omega_d^i x^{n/d}$$ occur also in the factorization (a consequence of Galois theory). These $d$ series are said conjugate, and are considered as a single branch of the curve, of ramification index $d$.

In the case of a real curve, that is a curve defined by a polynomial with real coefficients, three cases may occur. If none $P_i(x)$ has real coefficients, then one has a non-real branch. If some $P_i(x)$ has real coefficients, then one may choose it as $P_0(x)$. If $d$ is odd, then every real value of $x$ provides a real value of $P_0(x)$, and one has a real branch that looks regular, although it is singular if $d > 1$. If $d$ is even, then $P_0(x)$ and $P_{d/2}(x)$ have real values, but only for $x ≥ 0$. In this case, the real branch looks as a cusp (or is a cusp, depending on the definition of a cusp that is used).

For example, the ordinary cusp has only one branch. If it is defined by the equation $$y^2-x^3=0,$$ then the factorization is $$(y-x^{3/2})(y+x^{3/2});$$ the ramification index is 2, and the two factors are real and define each a half branch. If the cusp is rotated, it equation becomes $$y^3-x^2=0,$$ and the factorization is $$(y-x^{2/3})(y-j^2x^{2/3})(y-(j^2)^2x^{2/3}),$$ with $$j=(1+\sqrt{-3})/2$$ (the coefficient $(j^2)^2$ has not been simplified to $j$ for showing how the above definition of $P_i(x)$ is specialized). Here the ramification index is 3, and only one factor is real; this shows that, in the first case, the two factors must be considered as defining the same branch.

Non-plane algebraic curves
An algebraic curve is an algebraic variety of dimension one. This implies that an affine curve in an affine space of dimension n is defined by, at least, n − 1 polynomials in n variables. To define a curve, these polynomials must generate a prime ideal of Krull dimension 1. This condition is not easy to test in practice. Therefore, the following way to represent non-plane curves may be preferred.

Let $$f, g_0, g_3, \ldots, g_n$$ be n polynomials in two variables x1 and x2 such that f is irreducible. The points in the affine space of dimension n such whose coordinates satisfy the equations and inequations

$$\begin{align} &f(x_1,x_2)=0\\ &g_0(x_1,x_2)\neq 0\\ x_3&=\frac{g_3(x_1,x_2)}{g_0(x_1,x_2)}\\ & {}\ \vdots \\ x_n&=\frac{g_n(x_1,x_2)}{g_0(x_1,x_2)} \end{align}$$

are all the points of an algebraic curve in which a finite number of points have been removed. This curve is defined by a system of generators of the ideal of the polynomials h such that it exists an integer k such $$g_0^kh$$ belongs to the ideal generated by $$f, x_3g_0-g_3, \ldots, x_ng_0-g_n$$. This representation is a birational equivalence between the curve and the plane curve defined by f. Every algebraic curve may be represented in this way. However, a linear change of variables may be needed in order to make almost always injective the projection on the two first variables. When a change of variables is needed, almost every change is convenient, as soon as it is defined over an infinite field.

This representation allows us to deduce easily any property of a non-plane algebraic curve, including its graphical representation, from the corresponding property of its plane projection.

For a curve defined by its implicit equations, above representation of the curve may easily deduced from a Gröbner basis for a block ordering such that the block of the smaller variables is (x1, x2). The polynomial f is the unique polynomial in the base that depends only of x1 and x2. The fractions gi/g0 are obtained by choosing, for i = 3, ..., n, a polynomial in the basis that is linear in xi and depends only on x1, x2 and xi. If these choices are not possible, this means either that the equations define an algebraic set that is not a variety, or that the variety is not of dimension one, or that one must change of coordinates. The latter case occurs when f exists and is unique, and, for i = 3, …, n, there exist polynomials whose leading monomial depends only on x1, x2 and xi.

Algebraic function fields
The study of algebraic curves can be reduced to the study of irreducible algebraic curves: those curves that cannot be written as the union of two smaller curves. Up to birational equivalence, the irreducible curves over a field F are categorically equivalent to algebraic function fields in one variable over F. Such an algebraic function field is a field extension K of F that contains an element x which is transcendental over F, and such that K is a finite algebraic extension of F(x), which is the field of rational functions in the indeterminate x over F.

For example, consider the field C of complex numbers, over which we may define the field C(x) of rational functions in C. If $y^{2} = x^{3} − x − 1$, then the field C(x, y) is an elliptic function field. The element x is not uniquely determined; the field can also be regarded, for instance, as an extension of C(y). The algebraic curve corresponding to the function field is simply the set of points (x, y) in C2 satisfying $y^{2} = x^{3} − x − 1$.

If the field F is not algebraically closed, the point of view of function fields is a little more general than that of considering the locus of points, since we include, for instance, "curves" with no points on them. For example, if the base field F is the field R of real numbers, then $x^{2} + y^{2} = −1$ defines an algebraic extension field of R(x), but the corresponding curve considered as a subset of R2 has no points. The equation $x^{2} + y^{2} = −1$ does define an irreducible algebraic curve over R in the scheme sense (an integral, separated one-dimensional schemes of finite type over R). In this sense, the one-to-one correspondence between irreducible algebraic curves over F (up to birational equivalence) and algebraic function fields in one variable over F holds in general.

Two curves can be birationally equivalent (i.e. have isomorphic function fields) without being isomorphic as curves. The situation becomes easier when dealing with nonsingular curves, i.e. those that lack any singularities. Two nonsingular projective curves over a field are isomorphic if and only if their function fields are isomorphic.

Tsen's theorem is about the function field of an algebraic curve over an algebraically closed field.

Complex curves and real surfaces
A complex projective algebraic curve resides in n-dimensional complex projective space CPn. This has complex dimension n, but topological dimension, as a real manifold, 2n, and is compact, connected, and orientable. An algebraic curve over C likewise has topological dimension two; in other words, it is a surface.

The topological genus of this surface, that is the number of handles or donut holes, is equal to the geometric genus of the algebraic curve that may be computed by algebraic means. In short, if one consider a plane projection of a nonsingular curve that has degree d and only ordinary singularities (singularities of multiplicity two with distinct tangents), then the genus is $(d − 1)(d − 2)/2 − k$, where k is the number of these singularities.

Compact Riemann surfaces
A Riemann surface is a connected complex analytic manifold of one complex dimension, which makes it a connected real manifold of two dimensions. It is compact if it is compact as a topological space.

There is a triple equivalence of categories between the category of smooth irreducible projective algebraic curves over C (with non-constant regular maps as morphisms), the category of compact Riemann surfaces (with non-constant holomorphic maps as morphisms), and the opposite of the category of algebraic function fields in one variable over C (with field homomorphisms that fix C as morphisms). This means that in studying these three subjects we are in a sense studying one and the same thing. It allows complex analytic methods to be used in algebraic geometry, and algebraic-geometric methods in complex analysis and field-theoretic methods to be used in both. This is characteristic of a much wider class of problems in algebraic geometry.

See also algebraic geometry and analytic geometry for a more general theory.

Singularities
Using the intrinsic concept of tangent space, points P on an algebraic curve C are classified as smooth (synonymous: non-singular), or else singular. Given n − 1 homogeneous polynomials in n + 1 variables, we may find the Jacobian matrix as the (n − 1)×(n + 1) matrix of the partial derivatives. If the rank of this matrix is n − 1, then the polynomials define an algebraic curve (otherwise they define an algebraic variety of higher dimension). If the rank remains n − 1 when the Jacobian matrix is evaluated at a point P on the curve, then the point is a smooth or regular point; otherwise it is a singular point. In particular, if the curve is a plane projective algebraic curve, defined by a single homogeneous polynomial equation f(x,y,z) = 0, then the singular points are precisely the points P where the rank of the 1×(n + 1) matrix is zero, that is, where $$\frac{ \partial f }{ \partial x }(P)=\frac{ \partial f }{ \partial y }(P)=\frac{ \partial f }{ \partial z }(P)=0.$$

Since f is a polynomial, this definition is purely algebraic and makes no assumption about the nature of the field F, which in particular need not be the real or complex numbers. It should, of course, be recalled that (0,0,0) is not a point of the curve and hence not a singular point.

Similarly, for an affine algebraic curve defined by a single polynomial equation f(x,y) = 0, then the singular points are precisely the points P of the curve where the rank of the 1×n Jacobian matrix is zero, that is, where

$$f(P)=\frac{ \partial f }{ \partial x }(P)=\frac{ \partial f }{ \partial y }(P)=0.$$

The singularities of a curve are not birational invariants. However, locating and classifying the singularities of a curve is one way of computing the genus, which is a birational invariant. For this to work, we should consider the curve projectively and require F to be algebraically closed, so that all the singularities which belong to the curve are considered.

Classification of singularities
Singular points include multiple points where the curve crosses over itself, and also various types of cusp, for example that shown by the curve with equation x3 = y2 at (0,0).

A curve C has at most a finite number of singular points. If it has none, it can be called smooth or non-singular. Commonly, this definition is understood over an algebraically closed field and for a curve C in a projective space (i.e., complete in the sense of algebraic geometry). For example, the plane curve of equation $$y-x^3=0$$ is considered as singular, as having a singular point (a cusp) at infinity.

In the remainder of this section, one considers a plane curve $C$ defined as the zero set of a bivariate polynomial $f(x, y)$. Some of the results, but not all, may be generalized to non-plane curves.

The singular points are classified by means of several invariants. The multiplicity $m$ is defined as the maximum integer such that the derivatives of $f$ to all orders up to $m – 1$ vanish (also the minimal intersection number between the curve and a straight line at $P$). Intuitively, a singular point has delta invariant $δ$ if it concentrates $δ$ ordinary double points at $P$. To make this precise, the blow up process produces so-called infinitely near points, and summing $m(m − 1)/2$ over the infinitely near points, where m is their multiplicity, produces $δ$. For an irreducible and reduced curve and a point $P$ we can define $δ$ algebraically as the length of $$\widetilde{\mathcal{O}}_P / \mathcal{O}_P$$ where $$\mathcal{O}_P$$ is the local ring at P and $$\widetilde{\mathcal{O}}_P$$ is its integral closure.

The Milnor number $μ$ of a singularity is the degree of the mapping $grad f(x,y)⁄|grad f(x,y)|$ on the small sphere of radius ε, in the sense of the topological degree of a continuous mapping, where $grad f$ is the (complex) gradient vector field of f. It is related to δ and r by the Milnor–Jung formula,

Here, the branching number r of P is the number of locally irreducible branches at P. For example, r = 1 at an ordinary cusp, and r = 2 at an ordinary double point. The multiplicity m is at least r, and that P is singular if and only if m is at least 2. Moreover, δ is at least m(m-1)/2.

Computing the delta invariants of all of the singularities allows the genus g of the curve to be determined; if d is the degree, then

$$g = \frac{1}{2}(d-1)(d-2) - \sum_P \delta_P,$$

where the sum is taken over all singular points P of the complex projective plane curve. It is called the genus formula.

Assign the invariants [m, δ, r] to a singularity, where m is the multiplicity, δ is the delta-invariant, and r is the branching number. Then an ordinary cusp is a point with invariants [2,1,1] and an ordinary double point is a point with invariants [2,1,2], and an ordinary m-multiple point is a point with invariants [m, m(m − 1)/2, m].

Rational curves
A rational curve, also called a unicursal curve, is any curve which is birationally equivalent to a line, which we may take to be a projective line; accordingly, we may identify the function field of the curve with the field of rational functions in one indeterminate F(x). If F is algebraically closed, this is equivalent to a curve of genus zero; however, the field of all real algebraic functions defined on the real algebraic variety x2 + y2 = −1 is a field of genus zero which is not a rational function field.

Concretely, a rational curve embedded in an affine space of dimension n over F can be parameterized (except for isolated exceptional points) by means of n rational functions of a single parameter t; by reducing these rational functions to the same denominator, the n+1 resulting polynomials define a polynomial parametrization of the projective completion of the curve in the projective space. An example is the rational normal curve, where all these polynomials are monomials.

Any conic section defined over F with a rational point in F is a rational curve. It can be parameterized by drawing a line with slope t through the rational point, and an intersection with the plane quadratic curve; this gives a polynomial with F-rational coefficients and one F-rational root, hence the other root is F-rational (i.e., belongs to F) also.

For example, consider the ellipse x2 + xy + y2 = 1, where (−1, 0) is a rational point. Drawing a line with slope t from (−1,0), y = t(x + 1), substituting it in the equation of the ellipse, factoring, and solving for x, we obtain

$$x = \frac{1-t^2}{1+t+t^2}.$$

Then the equation for y is

$$y=t(x+1)=\frac{t(t+2)}{1+t+t^2}\,,$$

which defines a rational parameterization of the ellipse and hence shows the ellipse is a rational curve. All points of the ellipse are given, except for (−1,1), which corresponds to t = ∞; the entire curve is parameterized therefore by the real projective line.

Such a rational parameterization may be considered in the projective space by equating the first projective coordinates to the numerators of the parameterization and the last one to the common denominator. As the parameter is defined in a projective line, the polynomials in the parameter should be homogenized. For example, the projective parameterization of the above ellipse is

$$X=U^2-T^2,\quad Y=T\,(T+2\,U),\quad Z=T^2+TU+U^2.$$

Eliminating T and U between these equations we get again the projective equation of the ellipse $$X^2+X\,Y+Y^2=Z^2,$$ which may be easily obtained directly by homogenizing the above equation.

Many of the curves on Wikipedia's list of curves are rational and hence have similar rational parameterizations.

Rational plane curves
Rational plane curves are rational curves embedded into $$\mathbb{P}^2$$. Given generic sections $$s_1,s_2,s_3 \in \Gamma(\mathbb{P}^1, \mathcal{O}(d))$$ of degree $$d$$ homogeneous polynomials in two coordinates, $$x,y$$, there is a map$$s:\mathbb{P}^1 \to \mathbb{P}^2$$ given by $$s([x:y]) = [s_1([x:y]):s_2([x:y]):s_3([x:y])]$$defining a rational plane curve of degree $$d$$. There is an associated moduli space $$\mathcal{M} = \overline{\mathcal{M}}_{0,0}(\mathbb{P}^2, d\cdot [H])$$ (where $$[H]$$ is the hyperplane class) parametrizing all such stable curves. A dimension count can be made to determine the moduli spaces dimension: There are $$d+1$$ parameters in $$\Gamma(\mathbb{P}^1, \mathcal{O}(d))$$ giving $$3d+3$$ parameters total for each of the sections. Then, since they are considered up to a projective quotient in $$\mathbb{P}^2$$ there is $$1$$ less parameter in $$\mathcal{M}$$. Furthermore, there is a three dimensional group of automorphisms of $$\mathbb{P}^1$$, hence $$\mathcal{M}$$ has dimension $$3d + 3 - 1 - 3 = 3d - 1$$. This moduli space can be used to count the number $$N_d$$ of degree $$d$$ rational plane curves intersecting $$3d-1$$ points using Gromov–Witten theory. It is given by the recursive relation$$N_d = \sum_{d_A + d_B = d} N_{d_A} N_{d_B} d_A^2 d_B\left( d_B\binom{3d-4}{3d_A-2} - d_A\binom{3d-4}{3d_A-1} \right)$$where $$N_1 = N_2 = 1$$.

Elliptic curves
An elliptic curve may be defined as any curve of genus one with a rational point: a common model is a nonsingular cubic curve, which suffices to model any genus one curve. In this model the distinguished point is commonly taken to be an inflection point at infinity; this amounts to requiring that the curve can be written in Tate-Weierstrass form, which in its projective version is $$y^2z + a_1 xyz + a_3 yz^2 = x^3 + a_2 x^2z + a_4 xz^2 + a_6 z^3.$$

If the characteristic of the field is different from 2 and 3, then a linear change of coordinates allows putting $$a_1=a_2=a_3=0,$$ which gives the classical Weierstrass form $$y^2 = x^3 + p x + q.$$

Elliptic curves carry the structure of an abelian group with the distinguished point as the identity of the group law. In a plane cubic model three points sum to zero in the group if and only if they are collinear. For an elliptic curve defined over the complex numbers the group is isomorphic to the additive group of the complex plane modulo the period lattice of the corresponding elliptic functions.

The intersection of two quadric surfaces is, in general, a nonsingular curve of genus one and degree four, and thus an elliptic curve, if it has a rational point. In special cases, the intersection either may be a rational singular quartic or is decomposed in curves of smaller degrees which are not always distinct (either a cubic curve and a line, or two conics, or a conic and two lines, or four lines).

Curves of genus greater than one
Curves of genus greater than one differ markedly from both rational and elliptic curves. Such curves defined over the rational numbers, by Faltings's theorem, can have only a finite number of rational points, and they may be viewed as having a hyperbolic geometry structure. Examples are the hyperelliptic curves, the Klein quartic curve, and the Fermat curve $x^{n} + y^{n} = z^{n}$ when $n$ is greater than three. Also projective plane curves in $$\mathbb{P}^2$$ and curves in $$\mathbb{P}^1\times\mathbb{P}^1$$ provide many useful examples.

Projective plane curves
Plane curves $$C \subset \mathbb{P}^2$$ of degree $$k$$, which can be constructed as the vanishing locus of a generic section $$s \in \Gamma(\mathbb{P}^2, \mathcal{O}(k))$$, has genus $$\frac{(k-1)(k-2)}{2}$$ which can be computed using Coherent sheaf cohomology. Here's a brief summary of the curves genera relative to their degree For example, the curve $$x^4 + y^4 + z^4$$ defines a curve of genus $$3$$ which is smooth since the differentials $$4x^3, 4y^3, 4z^3$$ have no common zeros with the curve.. A non-example of a generic section is the curve $$x(x^2 + y^2 + z^2)$$ which, by Bezouts theorem, should intersect at most $$2$$ points, is the union of two rational curves $$C_1 \cup C_2$$ intersecting at two points. Note $$C_1$$ is given by the vanishing locus of $$x$$ and $$C_2$$ is given by the vanishing locus of $$x^2 + y^2 + z^2$$. These can be found explicitly: a point lies in both if $$x = 0$$. So the two solutions are the points $$[0:y:z]$$ such that $$y^2 + z^2 = 0$$, which are $$[0:1:-\sqrt{-1}]$$ and $$[0: 1: \sqrt{-1}]$$.

Curves in product of projective lines
Curve $$C \subset \mathbb{P}^1\times\mathbb{P}^1$$ given by the vanishing locus of $$s \in \Gamma(\mathbb{P}^1\times\mathbb{P}^1, \mathcal{O}(a,b))$$, for $$a,b \geq 2$$, give curves of genus$$ab - a -b + 1$$which can be checked using Coherent sheaf cohomology. If $$a = 2$$, then they define curves of genus $$2b -2 -b + 1 = b-1$$, hence a curve of any genus can be constructed as a curve in $$\mathbb{P}^1\times\mathbb{P}^1$$. Their genera can be summarized in the table and for $$a = 3$$, this is

Classical algebraic geometry

 * Acnode
 * Bézout's theorem
 * Cramer's theorem (algebraic curves)
 * Crunode
 * Curve
 * Curve sketching
 * Jacobian variety
 * Klein quartic
 * List of curves
 * Hilbert's sixteenth problem
 * Cubic plane curve
 * Hyperelliptic curve

Modern algebraic geometry

 * Birational geometry
 * Conic section
 * Elliptic curve
 * Fractional ideal
 * Function field of an algebraic variety
 * Function field (scheme theory)
 * Genus (mathematics)
 * Polynomial lemniscate
 * Quartic plane curve
 * Rational normal curve
 * Riemann–Roch theorem for algebraic curves
 * Weber's theorem (Algebraic curves)

Geometry of Riemann surfaces

 * Riemann–Hurwitz formula
 * Riemann–Roch theorem for Riemann surfaces
 * Riemann surface

title: Algebraic curve