Hatem Diaf
email: hatem.diaf@univ-lyon1.fr

António Pereira

Patrice Mélinon

Nicholas Blanchard

Florent Bourquard

Florence Garrelie

Christophe Donnet

Martin Vondráčk

Antonio Pereira

Patrice Melinon

Revisiting thin film of glassy carbon

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Amorphous carbon (a-C) is widely reported in the literature and used in many applications due to the large adaptability of its properties [1][2][3][4][5][6][7][8][9][10]. Its structure contains short range ordered domains mixing three hybridizations sp, sp 2 and sp 3 with different proportions of each them depending on deposition experimental parameters which in turn determines their properties. The particular case containing an important fraction of sp 3 is commonly known as diamond-like carbon (DLC) or tetrahedral carbon (ta-C) and it is mainly used for protective coatings in various scientific and industrial applications because of its highly mechanical and tribological properties [11].

Post-treatments of as-deposited carbon films are the subject of considerable interest. For example, Ferrari et al [12] showed that conventional thermal annealing under vacuum of ta-C thin films permits to efficiently modify its structure by converting sp 3 sites to sp 2 with a drastic change occurring after 1100 o C. More recently, Miyajima et al [10] studied the modification of electrical properties of a-C irradiated by pulsed UV laser and showed an increase in the sp 2 content with a reduction in resistivity. Such laser based approaches can be carried out in air without carbon oxidation (see Figure S1 Supporting Information), and it permits rapid heating and cooling with precise temporal control. Laser induced sp 3 to sp 2 transformation occurs therefore on the order of seconds compared to conventional annealing treatments which take hours [START_REF] Abrahamson | TechConnect Briefs 2018 -Materials for Energy, Efficiency and Sustainability[END_REF]. Moreover, the possibility to easily choose the temperature by adjusting the laser parameters and therefore the thermal diffusion depth in the film during the laser treatment permits the formation of SiC to be avoided and eases the integration of such materials with any kind of substrate. Many studies showed, on the basis of the Raman results, that laser irradiated amorphous carbon thin films have the structure of graphitic clusters or glassy carbon (GC, also called glass-like carbon or vitreous carbon) [1,[START_REF] Cappelli | [END_REF][15][16]. In particular, glassy carbon is attracting the interest of materials scientists and industrial companies for its large and original physical properties compared to other carbonaceous materials, such as high electrical conductivity, excellent mechanical resistance and gas impermeability. GC is also biologically compatible with living tissues [17][18][START_REF] Hassler | Coatings for biomedical applications[END_REF]. Finally, GC has the particularity of being the carbonaceous material with the lowest mass density (1.5 g/cm 3 ) [START_REF] Hassler | Coatings for biomedical applications[END_REF] without open porosity, thus combining the properties of vitreous and ceramic materials with those of graphite. For example, it is widely used as an electrode material in electrochemical applications, as well as for high temperature crucible [START_REF] Harris | [END_REF]. Glassy carbon is a sp 2 -hybridized, non-graphitisable and isotropic carbon with high structural order. According to the definition of International Union of Pure and Applied Chemistry (IUPAC), GC cannot be described as amorphous carbon. Harris [21][22][23][24] has described the glassy carbon as a fully sp 2 -bonded fullerene-related structure which consists of wrinkled fullerenes formed by curved graphene multilayers surrounding closed pores. The presence of this curvature is related to structural defects associated to non-hexagonal rings such as pentagons and heptagons [24]. GC is mainly produced by thermochemical decomposition of polymeric resins at high temperature (between 1000 and 2500 o C) [START_REF] Harris | [END_REF]25]. High temperature heating of precursor polymers results in an increase of the structural order within the layers as their size increases, as well as the appearance of numerous closed pores. This leads to a decrease in mass density to about 1.5 g/cm 3 and the formation of GC. In the final stage, at high temperature around 3000 o C, GC cannot change to crystalline graphite.

Its final microstructure depends on the chemical compositions and structure of the polymer precursors [26]. Even if this approach makes the formation of GC thin films possible, it suffers from many disadvantages: i) the need to work at very high temperatures, which reduces the number of possible substrates, ii) the adhesion to the substrate that can be affected by degassing during pyrolysis and also by the difference in the thermal expansion coefficient between the film and the substrate, and iii) the use of any precursor polymers to form GC causes the film to shrink during pyrolysis [27,28].

Herein, we report on a simple approach for the synthesis of glass-like carbon thin films at room temperature, which relies on the combination of two laser processes: pulsed laser deposition (PLD) and nanosecond UV laser irradiation. Thanks to an extensive characterization of the irradiated films, we show that after laser irradiation the deposited carbon film show all the features of glassy carbon for a carefully chosen set of process parameters: i) Raman spectra with the two main bands D and G which are situated at around 1330 cm -1 and 1590 cm -1 , respectively, together with a I D /I G ratio close to or higher than 1, and the weak T band at around 1050 cm -1 , ii) X-ray photoelectron (XPS) and Electron Energy loss spectroscopy (EELS) measurements show a high level of sp 2 hybridization, iii) a low mass density as determined by Rutherford backscattering (RBS), and iv) high resolution transmission electron microscopy (HTREM) images showing curved graphene multilayers surrounding closed pores. On the basis of this investigation, the formation of GC is theoretically discussed.

II. EXPERIMENTAL SECTION

Carbon thin films were deposited by PLD at room temperature by means of a pulsed KrF excimer laser (Lambda Physik, λ = 248 nm, pulse duration τ = 20 ns and repetition rate f = 10 Hz) and by using a commercially vitreous carbon (Neyco, 99.9 % purity, 1.54 g/cm 3 mass density, noted c.t.). Prior to deposition, the PLD chamber was pumped down to a residual pressure of ∼ 5.10 -7 mbar. The fluence was fixed to 12 J/cm 2 and the targetsubstrate distance was set to 4 cm. The carbon films were deposited on Si (100) substrate under vacuum (5.10 -7 mbar) or 1 mbar of He. The substrates were cleaned using ultrasound sonification and sequentially in acetone and ethanol and dried using dry nitrogen.

Pulsed laser irradiation of the deposited films was then performed in air using the same KrF excimer laser. Laser fluences were fixed to 0.15 J/cm 2 for carbon thin films deposited under He and 1.5 J/cm 2 for the one deposited under vacuum. The advantage of the use of UV laser is that the optical absorption depth is reduced and thereby avoids heating of the substrate. The thermal diffusion length is linked with the duration of the processing laser pulses [START_REF] Fontaine | Tribology of diamond-like carbon films[END_REF].

Raman spectra of carbon thin film were measured using an ARAMIS spectrometer (Horiba Jobin Yvon). A confocal mode was used to focus the laser beam on a sample surface with its displacement XYZ stage and a 100× objective of 0.95 numerical aperture. The experiments were carried out with the unpolarized excitation energy of 633 nm from a He-Ne laser. Spectral resolution of around 2 cm -1 was given by an air cooled CCD detector equipped with a grating of 600 lines.mm -1 . Prior to data acquisition, the calibration of the spectrometer was made by taking account of the line of the crystalline silicon at 520.5 cm -1 . A low laser energy density was used to irradiate the samples to avoid a laser induced change of its structure during the measurements. The Raman signals from lattice vibrations of carbonaceous materials are sensitive to the degree of graphitization and to structural disorder which are determined by the ratio between two bands I D and I G . The evolution of I D /I G can give information about structural changes. For all non-irradiated carbon thin films, only the D and G bands were fitted using Voigt functions, whereas for irradiated films and GC c.t., two components were added to be better fitted with the same functions: T peak ranging from 1120 to 1200 cm -1 and reported in [25,[START_REF] Sadezky | [END_REF], and A peak at around 1540 cm -1 originating from amorphous carbon coupound [25,[START_REF] Sadezky | [END_REF]. This analysis was supplemented by the HRTEM analysis performed using a JEOL JEM-2100 equipped with a High Tilt polepiece operated at 200 kV. The surface composition was studied by XPS using NanoESCA (Omicron Nanotechnology) equipped with a monochromatized Al Kα X-ray source (hν = 1486.7 eV). The pass energy of the analyzer was fixed at 100 eV. The samples were analyzed with a spot size of approximatively 100 µm. Fitting of the C 1s spectra were preformed using Voigt functions with Shirley background. When the asymmetry of the peak was high, Doniach-Sunjić peak shape convoluted with Gaussian function was used instead. The EELS measurements were carried out using a hemispherical electron energy analyzer (VG CLAM 4) operating with incident electron energies of 350 eV and 2 keV. The mass density of carbon thin film was obtained using 4 He + ions of 2 MeV energy delivered by the 4 MeV Van de Graaff accelerator. A detailed description of the setup was provided elsewhere [START_REF] Pelloux-Gervais | Caractérisation par méthodes nucléaires avancées de boîtes quantiques d'In (Ga) As épitaxiées sur silicium[END_REF].

III. RESULTS AND DISCUSSIONS A. Characterization of glassy carbon by Raman spectroscopy

According to the literature, the most popular technique used to characterize GC is Raman spectroscopy. The first-order Raman spectra of GC exhibit four main peaks between 1000 and 1800 cm -1 . Except for ultraviolet laser excitation, they are activated only by the sp 2 sites because of the resonant excitation of π states [START_REF] Ferrari | [END_REF]. The G band situated at around 1590 cm -1 is known as a feature of graphitic carbonaceous materials and is activated by a single resonant process. It corresponds to an optical phonon mode at the Γ Brillouin zone center with E 2g symmetry, i.e. the ordered sp 2 bonded carbon. In other words, all sp 2 bound carbon atoms within the layer (hexagonal rings and carbon chains) participate in the stretching mode vibration. The D band at around 1330 cm -1 is due to the breathing mode of sp 2 atoms in the six-ring carbon structure and is activated by structural disorder which doesn't exist in the perfectly ordered graphitic materials. The physical significance of the D mode was widely discussed and was finally assigned to the double-resonant Raman scattering associated to the TO phonon located near the Γ Brillouin zone [33]. The presence of structural defect induces the creation of holes in the bonds, promoting the double resonance scattering of single phonons by electron-hole relaxations [23]. Cançado et al. [34] showed the presence of armchairs and zigzag edges as active defects responsible for the double resonance Raman process and therefore for the increase in intensity of the D band. The primary characteristic feature of GC is the intensity ratio of the D and G bands (I D /I G ) which is expected to be greater than one. In other words, the D band corresponds to the presence of sixfold ring clusters in all nano-sized sp 2 domains surrounded by the defective edges. Ferrari and

Robertson [35] demonstrated that in disordered and amorphous carbons, the Tuinstra and Koening relation I D /I G ∼ 1/L a , which is valid for crystallite sizes higher than 2 nm, does not correspond to the development of the D band related to the growth of hexagonal rings and the increase in defects. They proposed a new relation

I D /I G ∝ L 2
a for defective carbons containing small crystallites below 2 nm. The D' band at about 1605 cm -1 is not easily observable and is due to the lattice vibration of graphene multilayers or small graphitic domains which are found in defective graphitic carbon. Thus, this band is also related to the presence of structural defects in sp 2 carbon materials such as D band but it is governed by intra-valley diffusion and not inter-valley diffusion as for D band. Finally, the T band, observed under UV excitation for tetrahedral amorphous carbon in the 1000-1300 cm -1 range, is due to the C-C sp 3 vibration [36]. But this position is also observable for GC under visible excitation and we assume that such T band could also originate from the vibrations of non-hexagonal ring carbon defects (i.e., heptagons, pentagons, octagons) [33,37].

B. Deposition of thin films by nanosecond pulsed laser ablation

As shown in Figure 1, the carbon film morphology is influenced by the nature of the atmosphere during deposition. As expected, under vacuum (Figure 1a), the film is smooth, uniform and compact with no appearance of porosity. For deposition under He atmosphere (Figure 1b), a porous film is clearly observed, with a cauliflower-like topology. With the increasing background pressure, a plume confinement and therefore a decrease in the velocity of the ablated species occurs, leading to the formation of a porous films [38,39]. This modification of carbon film morphology is in agreement with RBS measurements showing that the mass density of the film decreases from 1.9 g/cm 3 to 0.16 g/cm 3 when the back-ground pressure increases from vacuum to 1 mbar of He (Table I). Raman spectra of carbon thin films deposited by PLD as well as the spectrum of GC c.t. for comparison, are shown in Figure 2a-c. All the films deposited by PLD from GC c.t. exhibit amorphous carbon structure. The film deposited under vacuum (Figure 2b) shows spectrum with visible D peak less asymmetric than the one deposited under He (Figure 2c). The D and G intensity ratio is used as a tool to estimate the size of graphitic domains and is also used to obtain information on the carbon structure such as the degree of graphitization and the amount of disorder. To extract such quantitative information, the experimental spectra were fitted using Voigt functions for the D and G bands respectively. The results of the Raman spectra fitting (i.e., positions and full width at half maximum of D and G bands and the I D /I G ratio) are summarized in the Table I. The I D /I G ratios are similar for the porous and non-porous carbons films, indicating that they contains the same amount of hexagonal rings. Moreover the F W HM G and F W HM D values for porous and non-porous film are not significantly different. In conclusion, we assume that the effect of helium pressure is weak on the amorphous carbon structure but significant on the mass density [40].

C. Irradiation of thin film by UV nanosecond pulsed laser

The amorphous carbon layers were irradiated in air at the optimal fluence threshold (F th ) of 0.15 J/cm 2 and 1.5 J/cm 2 for the films deposited at 1 mbar of helium (porous film) and under vacuum (non-porous film), respectively (see Experimental section for more details).

These two different values of fluence are due to the difference in mass density of the asdeposited layers (see Table 1). Indeed, the denser the matter, the higher irradiation energy is needed to induce structural change. For a better comparison, we therefore determined the laser fluence allowing us to observe structural changes without damaging the layer ( up to around 200 pulses). Since the porous film (0.16 g/cm 3 ) is almost ten times less dense than the non-porous film (1.9 g/cm 3 ), it requires about ten times less laser energy. Figure 3 shows the evolution of the Raman spectra of porous carbon films with the number laser pulses.

More details on the data processing procedure are included in the Supporting Information (Figure S2). It can be noted that the D and G peaks become more distinct and separate as the number of laser pulses increases. The optimal Raman spectrum, which corresponds to the highest possible I D /I G ratio and the finest possible peaks, is obtained with 200 laser pulses. The Raman peak of silicon did not disappear with the sp 3 and sp 2 transition and its intensity remains with an increase in porosity. Above this value, the spectra remain the same except for an increase in peak intensity at 960 cm -1 (silicon substrate), indicating a decrease in the thickness of the carbon layer and therefore its ablation. A similar trend is observed for the non-porous film, for which an optimal Raman spectrum is obtained from 50 laser pulses (see Supporting Information, Figure S2). In this case, it is assumed that the disappearance of the silicon peak with sp 3 -sp 2 transition is due to a change in the optical properties of the non-porous layer. Similar observation can be made comparing graphite, which is completely sp 2 -bound and absorbs visible light, to diamond, which is sp 3 -bound and transparent. Figure 4 shows the evolution of the position and FWHM of D and G bands as well as the I D /I G ratio as a function of the number of laser pulses. As the number of laser pulses increases, the G band shifted to the higher wavenumbers at 1600 cm -1 whereas As visible Raman spectroscopy mainly provides information on the sp 2 component, it is not possible to evaluate the sp 2 /sp 3 ratio. However, such information can be obtained by XPS after deconvolution of the carbon peak and by calculating the peak area ratio corresponding to sp 2 and sp 3 components. High resolution XPS spectra in the C 1s region of as deposited and irradiated carbon films are shown in Figure 5b,c. For comparison, the XPS of the glassy carbon c.t. is also reported in Figure 5a. In all three cases, the C 1s spectrum displays two main components located at binding energy of 284.7 ± 0.2 and 285.5 ± 0.2 eV which are assigned to sp 2 (C=C bonds) known as the single feature of the perfect HOPG (Highly Oriented Pyrolytic Graphite) and sp 3 (C-C) hybridizations states seen as the signature of C-bonds in diamond, respectively. These values are consistent with the literature [41]. To obtain a perfect fit, additional components at 286.1 eV (C-O), 287.2 eV (C=O) and 289.0 eV (O-C=O) must be added. Therefore, the sp 3 hybridization state does not reflect the feature of GC widely viewed as fully sp 2 bonded carbon. This peak is in fact related to structural defect with distorted sp 2 orbitals (lattice distortion, vacancies and epoxy groups) [42][43][44][45][46]. For example, Barinov et al. [43] argued on the basis of density functional theory (DFT) calculations that the C 1s component between 285.3 and 285.9 eV should be added because of defective graphite. For the as-deposited porous carbon film at 1 mbar the results of the curve fitting show a mixture of sp 2 and sp 3 hybrid states with high fraction of sp 3 sites (68%) which is seen as diamond like carbon. The laser irradiation of the carbon film results in a significant change in its hybridization state, with a sp 2 rate that increases from around 32% to 72%. It can be noted that even if these results are in line with the Raman analyses, the sp 2 rate remains below that of the carbon c.t. The reason for this probably lies in the fact that this porous film has a higher surface which enhances the degree of defects, especially the chemisorption of oxygen atoms on the surface and consequently increases the defect peak. It should be noted that the defect peak does not decrease and remains unchanged even if the film started to be ablated, indicating that the deposited film is non-graphitisable (e.g., with 500 laser pulses increase of peak intensity peak at 960 cm -1 , see Figure 3). This modification of hybridization state with laser irradiation can be explained by the fact that sp 3 sites require less energy than sp 2 sites to be dissociated. As a result, sp 3 sites preferentially change to sp 2 leading to an ordered carbon structure. These results are supported by the EELS measurements (Figure 6) which indicates the presence of a π-plasmon peak at around 6 eV in the irradiated film. Glassy carbon, known as all sp 2 bonded carbon also exhibits this same peak located at loss energy of 6 eV corresponding to the collective excitation of π electrons (π -π * transitions) and indicating the sp 2 bonding structure [42,43]. The pulsed laser irradiation of the carbon film is accompanied by the bonding modification from amorphous carbon to ordered carbon before the beginning of ablation. We found the same binding change mechanism for both types of carbon films, namely porous and non-porous films. Nevertheless, while the ordered laser-induced structure seems identical in both cases, Figure 1 shows different macroscopic morphologies in these two films before laser irradiation. The SEM images in Figure 7 show the physical effect of laser irradiation on porous film (not visible on non-porous film due to its higher mass density, see Supporting Information Figure S3). It indicates that the laser irradiation induces the formation of holes inside the surface. Such morphological modification is accompanied by a decrease in the density of the layer. This decreases rapidly with the number of pulses to reach a constant value of about 0.09 g/cm 3 after 5 pulses (see Supporting Information, Figure S4). These results are in agreement with the works of Kononenko et al. [47] on laser-induced mechanisms in DLC films showing that the absorbed laser pulse produces an expansion of surface layer and reduces its mass density. Furthermore, Grigonis et al. [5] demonstrated that the conversion of the sp 3 to sp 2 sites caused the swelling.

Finally, the results presented so far show that pulsed laser irradiation of a porous and non-porous carbon film is accompanied by an increase in porosity and a modification of the bonds (conversion of the sp 3 to sp 2 sites), and leads to a better organized carbon structure.

Although all these characteristics tend to demonstrate the formation of GC, they may also correspond to other structures such as pyro-carbon. To validate the formation of GC, other characterization methods, such as low frequency Raman and HRTEM, were performed.

Low-frequency Raman analyses were mainly performed on non-porous films, due to the weak signal obtained with porous films (low mass density). However, the latter were used for HRTEM analyses. They are easy to exfoliate, which allows material to be deposited on a TEM grid without any mechanical thinning or ion beam for the preparation. The lowfrequency Raman spectra of non-porous films before and after laser irradiation are shown in Figure 8. The spectrum of the GC c.t. is also given for comparison. In the low-frequency region between 150 and 550 cm -1 , the two Raman peaks located at 224 and 334 cm -1 are signature of curvature-related structures which are normally present in GC. These peaks are also found for fullerenes, carbon nanotubes, or carbon onion structures, but completely absent in the case of graphite [48]. By now comparing the GC c.t. and the signs of irradiated film (Figure 8a and8b, respectively), it can be assumed that laser irradiation leads to the formation of heptagons and pentagons in the hexagonal lattice. As reported in the literature, [37,[49][50][51][52] these non-hexagonal rings induce the curvature in the network and make it isotropic. This structural distortion is observed in the HRTEM image of the irradiated porous-film (i.e., induced sp 2 -like carbon film, Figure 9b), where the stacking graphitic sheets are also visible unlike the amorphous structure before irradiation (Figure 9a). The GC c.t. (Figure 9c) and the sp 2 -like carbon film (Figure 9b) also show curved and facetted carbon sheets with two or three layers that contain voids of 5-10 nm in diameter. More in details, the same two structures (label 2) are wrapped by another graphene at a distance of about 0.32 nm. Its diameter is about 1.0 nm, which is identical to the size of a C 60 fullerene (label 1). Such fragments, observed in Figure 9b, contain less than 10 graphene sheets and cross in all directions and interconnect with each other. The average inter-layer distance extracted from Figure 9b is about 0.34 nm, which corresponds to turbostratic graphite [46]. The intensity profile of these multilayers fragments is shown in the Supporting Information (Figure S5). It is interesting to note that several particles (label 3) have a perfect hexagonal shape, i.e. well-defined facets in the shape of a hexagon with a 120 o curvature angle. Label 4 indicates two regions of negative (K < 0) and positive (K > 0) curvatures. Finally, the region labeled 5 shows the bifurcation of graphene layers indicating probably a local hyperbolic structure characterized by an inversion of the Gauss curvature (more details in the next section). These results clearly exhibit the strong similarity between glassy carbon and our irradiated carbon films (porous and non-porous). The laser-induced structural modification on the amorphous carbon film results in the formation of the isotropic sp 2 -like structure composed of the mixture of fullerene-like (K> 0), hyperbolic (K < 0) and curved ribbon-like structures which interconnect between them in all directions with the decrease in mass density. All results finally support that the sp 2 -like film is viewed as a glassy-like carbon structure.

IV. GLASS-LIKE (GL) CARBON STRUCTURE

The lack of long-range periodicity presents a theoretical challenge. As it is mentioned above, Raman features are not enough to fully characterize glassy carbon. For example, the same I D /I G ratio signature is reported in glassy graphene [54]. But there is one thing we can be sure of: GL is isotropic, porous, stable, metal-like and fully sp 2 hybridized. Because of the lack of long range order, ab initio calculations performed on large elemental cells are difficult and molecular dynamics based on semi classical potentials are elusive (see for example all the papers dedicated to carbon cluster structure [55,56]). However, simple geometrical considerations help us to qualitatively understand the structure. First of all, the sp 2 hybridized structure suggests a planar sheet which is in mathematics a surface in E 2 Euclidean space. A porous structure suggests a non-zero genus which is a main characteristic of surfaces in hyperbolic space (negative Gaussian curvature K). The isotropic structure of GL leads to a genus g ≥ 2 [57]. Hyperbolic space appears by the introduction of n-gons with n ≥ 7 (for example seven-membered or octogonal rings) in the primitive hexagonal structure (graphene) with the creation of saddle points (K < 0). Note that spherical space that characterize fullerenes appear by introducing five-membered rings instead of sevenmembered rings (genus in fullerenes is zero). The first prototype which meets all these features named "Schwarzite" was discussed a long time ago [58]. This hypothetic structure is based from triply periodic minimal surfaces (TPMS) (a minimal surface as a zero mean curvature radius) that ensure a cohesive energy not so far from other allotropic forms of carbon. TPMS structures are commonly deduced from differential geometry through the Weierstrass-Enneper relationship or approximated Fourier expansion. Despite the presence of n-gons n = 6 in the structure the strong sp 2 character is corroborated by the presence of Dirac cones as reported in graphene. A TPMS has no self-intersection and partitions the space into two disjoint labyrinthine regions. Contrary to fullerenes where isolated pentagon rule takes place, it seems that abutting heptagons are more stable [59]. Then, Schwarzites are probably less stable than disordered Schwarzites (abutting heptagons induce disorder in the structure). Experimental evidence of disordered Schwarzite structures was done by Barborini et al. [60], Rode et al. [61] and Wang et al. [62] n-fused rings with n = 6 in graphene introduces ripples and then a strain energy. Assuming the lowest strain energy, one has to consider five-membered, seven-membered rings or 8-membered rings no more.

Applying the Gauss-Bonnet theorem for the genus to a closed orientable surface, and Euler's law it holds

N 5 -N 7 -2N 8 = 12(1 -g) (1)
Non isotropic network with positive curvature writes

N 5 -N 7 -2N 8 = 12 (2)
Defect free fullerenes lead to N 5 = 12. If N 7 is positive, one observes fullerenes with (5,7)

Stone-Wales defects. Isotropic porous structures with g=3 yield:

N 5 -N 7 -2N 8 = 24 (3)
The simplest form with N 5 = N 8 = 0 gives the diamond D-type Schwarzite with only heptagons and 28 atoms per elemental cell (for one partition of space g = 2 [63]). Although TPMS is a theoretical object, our feeling is that porous isotropic structure is rich in heptagons (or n-gons with n > 7). The creation of (55,77) Stone-Wales defects in a pure graphene (formed by rotating a C-C bond by 90 o with regard to the midpoint of the bond) comes with a high energy cost (few eV) [50,52,64,65], but native pentagonal rings in the graphene structure considerably lowers the barrier for the heptagon formation up to an energy close to few k B T [64]. Since pentagonal rings are present in amorphous carbon with sp 2 /sp 3 hybridized states, the GL carbon structure can be formed if the basic structure before annealing is rich in sp 3 hybridized states. This is corroborated by experiments. Starting with rich graphite-like precursors is not the best way to prepare GL carbon. Conversely, the best that we can do is obtained from sp 2 /sp 3 amorphous precursors. Since infinite networks can be built up from basic TPMS structures, the choice of the candidate is arbitrary.

However, TPMS structures are either metallic or insulating [START_REF] Cataldo | The mathematics and topology of fullerenes[END_REF]. Experimental evidence of metal-like character after annealing suggests candidates with metal-like character. Amorphous TPMS structure can be obtained using a standard treatment of the primitive Fourier expansion with a scale distortion that preserves minimality according to Osseman's theorem [START_REF] Cataldo | The mathematics and topology of fullerenes[END_REF][START_REF] Osserman | A survey of minimal surfaces[END_REF]. The algorithm is decomposed in three steps (see reference [44]):

-Mathematical structure given by approximate Fourier expansion;

-Tessellation of the surface with hexagons and n-gons (n = 5, 7, 8) that mimics carbon structure with a crystallographic group space in E 3 Euclidean space;

-Randomization with Osserman's method.

Figure 10 displays such an example (gyroid structure with octamers) compared to 3D structure model of the porous 3D graphene using molecular mechanics method [53]. Even though this method relies on purely geometrical considerations, the obtained structure is qualitatively the same as other time-consuming computational methods.

V. CONCLUSION

Based on the data from all experimental results using two amorphous carbon film deposited in helium and in vacuum whose microstructures are similar at the nanometer scale but different at the macroscopic level, our in depth investigation shows that amorphous carbon with high sp 3 rate could be converted to glass-like carbon under ns-laser irradiation with the transition of sp 3 to sp 2 sites. sp 3 sites are seen as the precursor of the glassy-carbonlike formation. In addition, the Raman analysis at low frequency exhibits the formation of the Stone-Wales defects after laser annealing. So, laser irradiation involves two transformation mechanisms: creation of pentagons and heptagons in the hexagonal network and sp 3 -to-sp 2 phase change. When laser annealing is maintained without ablation for a long time, local graphitization degree increases with a decrease of defects. But as bonding is already established between graphitic fragments in the 3D space, the graphitization cannot be completely reached and some defects remain even at very high temperature, which gives non-graphitisable carbon such as glassy carbon. Moreover, laser irradiation also induces swelling in carbon during irradiation resulting in a decrease in mass density. The TEM observation indicates the coexistence between fullerenes and graphitic nano-crystallites which interconnect each other by curved sheets of graphene. Such results emphasize the strong resemblance between the microstructure of irradiated carbon film with that of glassy carbon. In conclusion, the irradiated carbon film obtained under theses experimental conditions change to a glassy carbon like one. It is now possible to synthetize a thin film of glassy carbon with only a laser for deposition and irradiation in place of the pyrolysis. This in-depth characterization offers to researchers some possibilities to use the thin film of glass carbon for their research activity and to material engineers for the next-generation applications which benefit from the properties of glassy carbon.

FIG. 2 .

 2 FIG. 2. Raman spectra of the glassy carbon c.t. (a) and of carbon thin films deposited by PLD at two different pressures: (b) 10 -6 mbar and (c) 1 mbar of helium. The peak at 950 cm -1 is due to the second order Raman band from the Si substrate. The red solid line is the result of the fit.

FIG. 3 .

 3 FIG. 3. Corrected Raman spectra of irradiated porous carbon film with different number of laser pulses : (a) un-irradiated carbon, (b) 5 shots, (c) 10 shots, (d) 100 shots, (e) 200 shots, (f) 1000 shots. See Supporting Information [Figure S2] for Raman spectra treatment.

FIG. 4 .

 4 FIG. 4. Characteristics (position and FWHM) of the G (a) and D (b) bands versus the number of laser pulses delivered on porous carbon film deposited by PLD at 1 mbar of He. The I D /I G ratio is given in (c). For the GC c.t., the two bands D and G are situated at around 1330 cm -1 and 1599 cm -1 respectively, and the I D /I G ratio is 2.19.

FIG. 5 .

 5 FIG. 5. High resolution XPS spectra of the C 1s peak: (a) glassy carbon c.t., (b) as-deposited carbon film (1 mbar of He), and (c) irradiated film with 1000 laser pulses.

FIG. 6 .

 6 FIG. 6. Electron energy loss spectra of as deposited carbon film (1mbar of He) and irradiated film with 200 laser pulses. Spectrum of the glassy carbon c.t. is also shown for comparison.

FIG. 7 .

 7 FIG. 7. SEM images of a porous carbon film (deposited at 1 mbar of helium) before (a) and after (b) laser irradiation.

FIG. 9 .

 9 FIG. 9. HRTEM images of porous carbon film: (a) before irradiation and (b) after irradiation with 200 pulses. (c) corresponds to the glassy carbon c.t.. The labels in (b) correspond to (1) fullerene, (2) multilayers fragment, (3) honeycomb-like fragment, (4) presence of curvature with the angle of 120 and (5) hyperbolic structure.

TABLE I .

 I Deconvoluted Raman data for the different films. GC(Glassy carbon c.t.),NP (Non porous film at 10 -6 mbar),P (Porous film/1mbar He),MD(Mass density) , λ D,G and F W HM D,G in cm -1 , MD in g/cm 3 .

		λ D	F W HM D	λ G	F W HM G	I D /I G	MD
	GC	1332	54	1600	64	2.2	1.5
	NP	1357	362	1533	213	0.9	1.9
	P	1350	365	1530	203	0.9	0.16

ACKNOWLEDGMENTS

We thank Olivier Boisron for EELS measurements. Photoemission experiments were performed at the NanoESCA facilities supported by MEYS of the Czech Republic through project SOLID21CZ.02.1.01/0.0/0.0/16 019/0000760


Giulio Caviglia
email: gcavigli@purdue.edu

Marc Chardin
email: marc.chardin@imj-prg.fr

Jason Mccullough

Irena Peeva

Matteo Varbaro
email: varbaro@dima.unige.it

Regularity of prime ideals

Keywords: 2010 Mathematics Subject Classification. Primary: 13D02 Syzygies, Free Resolutions, Castelnuovo-Mumford Regularity

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Introduction

Regularity is a numerical invariant that measures the complexity of the structure of homogeneous ideals in a polynomial ring. It has been studied in Algebraic Geometry and Commutative Algebra; see the expository paper [Ch]. We consider a standard graded polynomial ring U = k[z 1 , . . . , z p ] over a field k, where all variables have degree one. Let L be a homogeneous ideal in the ring U , and let β ij (L) = dim k Tor U i (L, k) j be its graded Betti numbers. The (Castelnuovo-Mumford) regularity of L is reg(L) = max j β i, i+j (L) = 0 .

Alternatively, regularity can be defined using local cohomology. Papers of Bayer-Mumford, Bayer-Stillman, and Koh, give examples of families of ideals attaining doubly exponential regularity. In contrast, Bertram-Ein-Lazarsfeld, Chardin-Ulrich, and Mumford have proved that there are nice bounds on the regularity of the ideals of smooth (or nearly smooth) projective varieties; see the expository paper [START_REF] Chardin | Bounds for Castelnuovo-Mumford regularity in terms of degrees of defining equations, Commutative algebra, singularities and computer algebra[END_REF]. As discussed in the influential paper [BM] by [START_REF] Bayer | What can be computed in Algebraic Geometry?[END_REF], the biggest missing link between the general case and the smooth case is to obtain a decent bound on the regularity of all prime ideals (the ideals that define irreducible projective varieties). The long standing Eisenbud-Goto Regularity Conjecture predicts an elegant linear bound, in terms of the degree of the variety:

The Regularity Conjecture 1.1. (Eisenbud-Goto [EG], 1984) Suppose that the field k is algebraically closed. If L ⊂ (z 1 , . . . , z p ) 2 is a homogeneous prime ideal in U , then

(1.1) reg(L) ≤ deg(U/L) -codim(L) + 1 ,
where deg(U/L) is the multiplicity of U/L (also called the degree of U/L, or the degree of X = Proj(U/L)), and codim(L) is the codimension (also called height) of L.

The conjecture is proved for curves by Gruson-Lazarsfeld-Peskine, for smooth surfaces by Lazarsfeld and Pinkham, for most smooth 3-folds by Ran and Kwak, if U/L is Cohen-Macaulay by Eisenbud-Goto, and in many other special cases.

Recently, McCullough and Peeva [MP] introduced two new techniques and used them to provide many counterexamples to the Eisenbud-Goto Regularity Conjecture. In this note we answer some natural questions which arise from the paper [MP].

The counterexamples in [MP] come from Rees-like algebras, which were introduced in [MP, Section 3]. Rees-like algebras, unlike the usual Rees algebras, have well-structured defining equations and minimal free resolutions. The properties of Rees algebras are of high interest and can be quite intricate (see for example [Hu], [KPU]). Several mathematicians have asked us if the defining ideals of Rees algebras contain counterexamples as well or whether the Regularity Conjecture holds for them. In Sections 3 and 4 we provide counterexamples using Rees algebras. In the latter section we study standard graded Rees algebras that arise as Rees algebras of ideals generated in one degree.

The main theorem in [MP] shows that the regularity of prime ideals is not bounded by any polynomial function of the multiplicity. It is natural to ask if there exists a bound on regularity in terms of the multiplicity alone. Such a bound does not exist for primary ideals (Example 5.4). However, we prove in Section 5 that the recent work of Ananyan-Hochster [AH] (who solved Stillman's Conjecture) implies the existence of the desired bound for prime ideals. Other proofs of Stillman's Conjecture are given in [ESS, DLL].

In the counterexamples in [MP] the multiplicity is smaller than the maximal degree of a minimal generator of a prime ideal. One may wonder whether there are prime ideals for which the difference between the maximal degree of a minimal generator and the maximal degree of a minimal (first) syzygy can be made arbitrarily large. In Section 6 we show that such prime ideals exist. We obtain them by starting with Ullery's designer ideals (which are not prime) [Ul] and applying to them the method by McCullough-Peeva in order to get prime ideals.

In Section 7 we construct a family of three-generated ideals whose regularity grows faster than the product of the degrees of the generators. To our knowledge, this is the only known such family other than those based on the Mayr-Meyer construction. Applying the construction in [MP] we use this family to construct an infinite family of counterexamples to the Eisenbud-Goto Regularity Conjecture that do not rely on the Mayr-Meyer construction.

Multiplicity of prime ideals

Throughout this section, we consider a polynomial ring W = k[w 1 , . . . , w p ] over an arbitrary field k and positively graded with deg(w i ) ∈ N for every i. Suppose c i := deg(w i ) > 1 for i ≤ q and deg(w i ) = 1 for i > q (for some q ≤ p).

A function

Q : Z -→ Q is a quasipolynomial (over Q) of degree r if Q(n) = a r (n)n r + a r-1 (n)n r-1 + • • • + a 1 (n)n + a 0 (n),
where a i : Z -→ Q is a periodic function for each i = 0, . . . , r and a r = 0. A natural number v is called a period of Q if

a i (n + v) = a i (n)
for all n ∈ Z and for all i = 0, . . . , r.

Let M be a homogeneous ideal in the polynomial ring W . The Hilbert function h W/M :

Z ≥0 -→ Z ≥0 of W/M is h W/M (n) = dim k (W/M ) n .
It is often studied via the Hilbert series

Hilb W/M (u) = n≥0 u n dim k (W/M ) n .
By a theorem of Hilbert-Serre, there is a quasipolynomial Q(n) of degree dim(W/M ) -1 and period lcm(c 1 , . . . , c p ) such that

h W/M (n) = Q(n) for n 0.
For a proof, see for example [BI]. Set d := gcd(c 1 , . . . , c p ), and observe that Q(dj + t) = 0 for 0 < t < d.

Proposition 2.1. If M is a prime ideal, then a r (dj) is a constant (independent of the parameter j), which we denote a r .

Proof. We may easily reduce to the case d = 1 by dividing the degrees of the variables by their greatest common divisor. Assume the opposite. Set a(n) = a r (n). Let m and m + s be two different integers for which the Hilbert function agrees with the quasipolynomial Q and such that a(m) > a(m + s). Since gcd(c 1 , . . . , c p ) = 1, there exist i ∈ Z such that s = q 1 i c i . Hence, m + s = m + q 1 i c i . Adding a large positive multiple of b := lcm(c 1 , . . . , c p ) to the righthand-side, we get

a(m + s) = a m + q i=1 i c i + vb = a m + q i=1 i c i
where each i is positive. Foir each i, as w i is a non-zerodivisor, we have an inclusion w i (W/M ) j ⊆ (W/M ) j+c i and thus dim k (W/M ) j ≤ dim k (W/M ) j+c i for every j ≥ 0. Hence,

a(m + s) = a m + q i=1 i c i ≥ a(m) ,
which is a contradiction.

If M is a prime ideal in W , then we call e Hilb (M ) := r!a r the Hilbert multiplicity of S/M or of M , and also denote it by deg(M ) or e Hilb (W/M ).

On the other hand, recall the construction of the Euler polynomial: Notation 2.2. Fix a finite graded complex V of finitely generated W -free modules and with V i = 0 for i < 0. We may write

V i = ⊕ j∈Z W (-j) b ij . The Euler polynomial of V is E V = i≥0 j∈Z (-1) i b ij u j .
Let N be a graded finitely generated W -module, and let V be a finite graded free resolution of N . Since every graded free resolution of N is isomorphic to the direct sum of the minimal graded free resolution and a trivial complex, it follows that the Euler polynomial does not depend on the choice of the resolution, so we call it the Euler polynomial of N . We factor out a maximal possible power of 1 -u and write

E V = (1 -u) c h V (u) , where h V (1) = 0.
We set N = W/M and in the notation above, we call e Euler (M ) := h V (1) the Euler multiplicity of S/M or of M , and also denote it by e Euler (W/M ).

A prime ideal M is called non-degenerate if M ⊂ (w 1 , . . . , w p ) 2 .

Theorem 2.3. If M is a non-degenerate homogeneous prime ideal in W , then

e Euler (M ) = e Hilb (M )

p i=1 deg(w i ) .
The proof uses the technique of step-by-step homogenization introduced in [MP]. Theorem 2.3 is an immediate corollary of Theorem 2.5.

The following result from [MP] describes the step-by-step homogenization technique:

Step-by-step Homogenization Theorem 2.4. [MP] Let M be a homogeneous nondegenerate prime ideal, and let K be a minimal set of homogeneous generators of M . Consider the homogenous map (of degree 0)

ν : W = k[w 1 , . . . , w p ] -→ W := k[w 1 , . . . , w p , v 1 , . . . , v q ] w i -→ w i v deg W (w i )-1 i for 1 ≤ i ≤ q ,
where v 1 , . . . , v q are new variables and W is standard graded. The ideal M ⊂ W generated by the elements of ν(K) is a homogeneous non-degenerate prime ideal in W . Furthermore, the graded Betti numbers of W /M over W are the same as those of W/M over W .

We say that M is obtained from M by step-by-step homogenization or by relabeling. The relation between the multiplicities of M and M is given in the next result: Proof. The first equality holds because M is a homogeneous ideal in a standard graded ring. The second equality holds because step-by-step homogenization preserves the graded Betti numbers by Theorem 2.4. We will prove the third equality.

We have the same Euler polynomial E(u) := E M = E M by Theorem 2.4. We get the Hilbert series

Hilb W/M (u) = E(u) (1 -u) p-q q i=1 1 -u deg(w i ) Hilb W /M (u) = E(u) (1 -u) p+q .
Therefore,

Hilb W /M (u) = q i=1 1 -u deg(w i ) (1 -u) 2q Hilb W/M (u) = q i=1 1 + u + • • • + u deg(w i )-1 (1 -u) q Hilb W/M (u) .
Note that the factor in front of Hilb W/M (u) is a series with positive coefficients. Apply Lemma 2.6.

If S/M has Hilbert multiplicity a, we say that its Hilbert series has it too.

Lemma 2.6. Let h be a Hilbert function given by a quasipolynomial Q(n) of degree r and with constant leading coefficient a := r!a r . Let g be the Hilbert series of h.

(1) The Hilbert multiplicity of the Hilbert series g 1 -u is the same.

(2) The Hilbert multiplicity of the Hilbert series

1 + u + • • • + u b g is (b + 1)a. Proof. We have g = n≥0 Q(n)u n .
(1) Since

(1 + u + u 2 + • • • )g = n≥0 u n Q(0) + Q(1) + • • • + Q(n)
the quasipolynomial for the considered Hilbert series is

Q(n) + • • • + Q(0) = a r n r + (n -1) r + • • • + 1 + a r-1 n r-1 + (n -1) r-1 + • • • + 1 + • • • = a r r + 1 n r+1 + terms of lower degree .
Thus the Hilbert multiplicity is (r + 1)

! a r r+1 = r!a r . (2) Since (1 + u + • • • + u b )g = n≥0 u n Q(n) + • • • + Q(n -b)
the quasipolynomial for the considered Hilbert series is

Q(n) + • • • + Q(n -b) = a r n r + (n -1) r + • • • + (n -b) r + a r-1 n r-1 + (n -1) r-1 + • • • + (n -b) r-1 + • • • = a r (b + 1)n r + terms of lower degree .

Multiplicities of Rees algebras and Rees-like algebras

In this section we provide counterexamples to the Eisenbud-Goto Regularity Conjecture [EG] using Rees algebras.

Notation 3.1. We follow the notation in [MP]. Consider the polynomial ring

S = k[x 1 , . . . , x n ]
over a field k with a standard grading defined by deg(x i ) = 1 for every i. Let I be a homogeneous ideal minimally generated by forms f 1 , . . . , f m of degrees a 1 , . . . , a m , where m ≥ 2.

Theorem 3.2. Consider the Rees algebra S[It] and the Rees-like algebra S[It, t 2 ]. We have:

e Hilb (S[It]) ≤ e Hilb (S[It, t 2 ]) = 1 e Euler (S[It]) ≤ 1 2 e Euler (S[It, t 2 ]) = m i=1 (deg(f i ) + 1) .
Denote by Q and L the defining ideals of S[It, t 2 ] and S[It] respectively. Then

deg L ≤ 1 2 deg Q ,
where L and Q (Q is denoted by P in [MP]) are the respective step-by-step homogenizations of L and Q. The first and second equalities below come from Theorem 2.3:

e Euler (S[It]) = e Hilb (S[It]) m i=1 deg(y i ) ≤ e Hilb (S[It, t 2 ]) m i=1 deg(y i ) = 1 2 e Euler (S[It, t 2 ]) = 1 2 e Euler (Q ) = m i=1 (deg(f i ) + 1) .
The factor 1 2 comes from the variable z which has degree 2. The last equality holds by [START_REF] Mccullough | Counterexamples to the Eisenbud-Goto Regularity Conjecture[END_REF]Theorem 1.6(2)], and the equality before holds by Theorem 2.5.

The inequality deg L ≤ 1 2 deg Q now follows from Theorem 2.5. For a graded ideal N (in a positively graded polynomial ring), we denote by maxdeg(N ) the maximal degree of an element in a minimal system of homogeneous generators of N .

Theorem 3.3. For r ∈ N we consider the step-by-step homogenization L r of the defining ideal L r of the Rees algebra S[I r t], where I r is the Koh ideal used in Counterexample 1.8(1) in [MP]. Then multiplicity and maxdeg of the prime ideal L r satisfy

deg L r ≤ 2 × 3 22r-3 maxdeg L r ≥ 2 2 r-1 + 1 .
Thus it is a counterexample to the Regularity Conjecture (see [START_REF] Mccullough | Counterexamples to the Eisenbud-Goto Regularity Conjecture[END_REF]1.2]) for r ≥ 10.

Proof. Note that L r contains all the y-linear minimal generators listed in [START_REF] Mccullough | Counterexamples to the Eisenbud-Goto Regularity Conjecture[END_REF](3.4)]. They are minimal generators of L r by [MP, Proposition 2.9] and since L r cannot contain any elements in which no y 1 , . . . , y m appears. Let P r be the prime ideal used in Counterexample 1.8(1) in [MP]. Then

deg L r ≤ 1 2 deg P r ≤ 2 × 3 22r-3
by Theorem 3.2 and [START_REF] Mccullough | Counterexamples to the Eisenbud-Goto Regularity Conjecture[END_REF]Counterexample 1.8(1)].

Similarly, Counterexample 1.8(2) in [MP] leads to Rees-algebra counterexamples to the Regularity Conjecture.

Standard graded Rees Algebras

In this section, we provide a different view than Theorem 3.3 on using Rees Algebras to produce examples of large regularity. We focus on standard graded Rees Algebras, which arise as the Rees algebras of ideals generated in one degree.

First, we observe how to reduce to the case of ideals generated in one degree:

Construction 4.1. We follow the notation in [MP]. We bigrade S by deg(x i ) = (1, 1) for every i and deg(x) = (0, 1). The ideal I is bigraded, and therefore S/ I has a bigraded minimal free resolution U over S. The regularity r of S/ I (assuming standard grading) is equal to the regularity of U with respect to the second coordinate of the bigrading. It follows that it is bigger than the regularity r of U with respect to the first coordinate of the bigrading since deg(x) = (0, 1). Observe that x -1 is a non-zerodivisor (for degree reasons using the second coordinate of the bigrading) on S/ I. Therefore, U ⊗ S/(x -1) is a graded (posibly non-minimal) free resolution of S/I over S.

Hence, the regularity r of S/I is smaller than r . We showed that reg S (I) ≤ reg S ( S/ I) .

In fact, we have such an inequality in every homological degree, that is, maxdeg(Syz S i (S/I)) ≤ maxdeg(Syz S i ( S/ I)) ,

where maxdeg(N ) stands for the maximal degree in a system of minimal homogeneous generators of a graded finitely generated module N . Now, we consider the Rees Algebra R := S[ It] as a standard graded ring. Its prime graded (with respect to the standard grading) defining ideal T satisfies

maxdeg(T ) ≥ maxdeg(Syz S 1 ( I)) -(d -1) ≥ maxdeg(Syz S 1 (I)) -(d -1) .
Example 4.2. We will apply Construction 4.1 to Koh's examples based on the Mayr-Meyer [MM] construction. For r ≥ 1, Koh constructed in [Ko] an ideal I r generated by 22r -3 quadrics and one linear form in a polynomial ring with 22r -1 variables, and such that maxdeg(Syz 1 (I r )) ≥ 2 2 r-1 . The construction above produces an ideal I r generated by 22r-2 quadrics in a polynomial ring with 22r variables, and such that maxdeg(T r ) ≥ 2 2 r-1 -1. On the other hand, by Theorem 4.3,

deg(T r ) ≤ 2 min{22r-2, 22r} -1 = 2 22r-2 -1 .
Thus, deg(T r ) < maxdeg(T r ) for r ≥ 10. Now, we turn to Rees Algebras.

deg(R M ) ≤ d min{m,n} -1 d -1 ,
and equality holds if further M is (X 1 , . . . , X n )-primary or its m generators form a regular sequence.

Proof. Call g 1 , . . . , g m the given m generators of M . First give bidegree (1, 0) to the X i 's and bidegree (0, 1) to the Y i 's.

Notice that each bigraded component (R M ) p,j for this bigrading has at most a vector space dimension equal to the one of the Rees algebra associated to generic forms (ones with indeterminate coefficients). Indeed, this dimension can be computed as the rank of a Sylvester matrix associated to the collection of elements

g i 1 • • • g i j with i 1 ≤ • • • ≤ i j in the degree p + jd.
This inequality in turn shows that the Hilbert function of R M is bounded above by the one given by generic forms (over the extension of k generated by the coefficients).

If m ≤ n the generic forms are providing a regular sequence. And any complete intersection has its Rees algebra resolved by the Eagon-Northcott complex of the (2 × m)-matrix with maximal minors Y s g r -Y r g s .

Let Hilb R M (u) be the Hilbert series of R M . Whenever m ≤ n this shows the (term by term) inequality :

Hilb R M (u) ≤ (1 -u d ) m-1 + u d m-2 i=0 (1 -u d ) i (1 -u) m-i-1 (1 -u) m+n .
As M = 0, the dimension of the Rees algebra of M is the same as the one of a complete intersection and hence the inequality above shows that the degree of R M is bounded above by the one corresponding to a complete intersection, whose value is d m-1 +d m-2 +• • •+d+1 by the above formula for the complete intersection case. Notice further that any graded ideal generated in degree d is a subideal of J := (X 1 , . . . , X n ) d . It follows that the Hilbert series of R M is bounded above by the one of R J . Again, recall that R M and R J have same dimension, hence

deg(R M ) ≤ deg(R J ) = d n -1 d -1 .
We computed the multiplicity in the complete intersection case, and if M is (X 1 , . . . , X n )primary, it contains an ideal M generated by a regular sequence of forms of degree d. The degree of R M is hence bounded below by the degree of R M and above by the one R J ; these are both equal to d n -1 d-1 and the conclusion follows.

Regularity is bounded in terms of multiplicity

In this section we show that an upper bound on regularity of non-degenerate prime ideals in terms of the multiplicity alone follows from the recent work of Ananyan and Hochster [AH], who solved Stillman's Conjecture. From now on, the polynomial rings occurring in the paper are standard graded.

Lemma 5.1. Let L 1 and L 2 be two homogeneous ideals of U = k[X 1 , . . . , X N ] whose number and degrees of generators are bounded by a constant c. Then the number and the degrees of the generators of L 1 ∩ L 2 are bounded by a constant depending only on c.

Proof. Looking at the exact sequence

0 -→ U L 1 ∩ L 2 -→ U L 1 ⊕ U L 2 -→ U L 1 + L 2 -→ 0
we infer the following inequality for any j ∈ Z:

dim k Tor U 0 U L 1 ∩ L 2 , k j ≤ dim k Tor U 0 U L 1 , k j + dim k Tor U 0 U L 2 , k j + dim k Tor U 1 U L 1 + L 2 , k j .
By [AH, Theorem D (a)] the regularity and the graded Betti numbers of U L 1 +L 2 are bounded by a constant depending only on c, so we get the desired property.

Theorem 5.2. Let e and h be positive integers and k be a field. There exist constants, depending only on e and h, bounding respectively the projective dimension, regularity, and the graded Betti numbers of every homogeneous unmixed radical ideal of multiplicity e and height h in a standard graded polynomial ring over k.

Proof. We may assume e ≥ 2. Let L be a homogeneous unmixed radical ideal of U = k[X 1 , . . . , X N ] of multiplicity e and height h.

First, we will show that the ideal L contains a regular sequence g 1 , . . . , g h of forms of degrees less than or equal to e. Choose a Noether normalization k[X 1 , . . . , X d ] (this may need a finite extension of the base field to change coordinates if |k| ≤ e, but this extension keeps L unmixed and radical and does not affect the invariants we are bounding). Then the generators of L ∩ k[X 1 , . . . , X d , X d+i ] = (g i ) for i = 1, . . . h form a regular sequence of forms of degrees at most e.

Let b := (g 1 , . . . , g h ) and

m := h i=1 (deg g i -1) = reg(U/b) .
If L = b the assertion is clear, so suppose L = b. If p i is a minimal prime of b, then there exists a form f i of degree m such that p i = b : (f i ) (by [START_REF] Chardin | Applications of some properties of the canonical module in computational projective algebraic geometry, Symbolic computation in algebra, analysis, and geometry[END_REF]4.1] or [START_REF] Chardin | Liaison and Castelnuovo-Mumford regularity[END_REF]1.2], for instance).

Hence, if L = ∩ a i=1 p i , then L = b : (f ) with f := a i=1 f i . The exact sequence 0 -→ U/L(-m) -→ U/b -→ U/(b + (f )) -→ 0 then shows that reg(U/L) = reg U/(b + (f )) -m + 1. Hence the regularity of L is bounded by the one of an ideal generated in degrees (e, . . . , e, m) such that e is repeated h times, and m ≤ h(e -1).

By [AH, Theorem D (a)] it follows that the projective dimension, regularity, and Betti numbers of L are bounded as well by constants depending only on e and h.

Corollary 5.3. Let e be a positive integer and k be an algebraically closed field. There exist constants, depending only on e, bounding the projective dimension, regularity and graded Betti numbers of every homogeneous non-degenerate prime ideal in a polynomial ring over k of multiplicity e.

Proof. The claim follows immediately by Theorem 5.2 since the height of a homogeneous non-degenerate prime ideal in a polynomial ring over an algebraically closed field is less than its multiplicity. This corollary also holds for reduced, equidimensionnal ideals that are connected in codimension 1 (or connected in codimension r, for any given r ≥ 1). Also the conclusion concerning the regularity holds without the condition of being non-degenerate.

Examples 5.4. We give three examples. They show that Corollary 5.3 cannot be generalized to radical ideals, nor to primary ideals.

(1) Fix n ∈ N and let

M = (x 1 , . . . , x n ) ∩ (y 1 , . . . , y n ) ⊆ U = k[x i , y i | i = 1, . . . , n] .
Then M is a non-degenerate unmixed radical ideal of multiplicity 2, but projdim U/M = 2n -1 .

(2) Fix n ∈ N and let

M = (x 2 , xy, y 2 , xa n + yb n ) ⊆ U = k[x, y, a, b] .
It follows from [START_REF] Engheta | On the projective dimension and the unmixed part of three cubics[END_REF]Lemma 10] that (i) M is a non-degenerate ideal of regularity equal to n + 1; (ii) e(U/M ) = 2; (iii) M is (x, y)-primary.

(3) Fix n, e ∈ N with e ≥ 3. By [START_REF] Huneke | Multiple structures with arbitrarily large projective dimension supported on linear spaces[END_REF]Theorem 1.2] there exists an ideal M in a polynomial ring U over k such that:

(i) M is a non-degenerate ideal of projective dimension at least n;

(ii) e(U/M ) = e; (iii) M is (x, y)-primary, where x and y are independent linear forms.

These examples still leave open the question whether there exists a bound on the regularity of unmixed radical ideals (over an algebraically closed field) in terms of the multiplicity alone.

Remarks 5.5. We provide three upper bounds on regularity. Use the notation in the introduction.

(1) According to a lemma which Mumford attributes to Castelnuovo [START_REF] Mumford | Lectures on curves on an algebraic surface[END_REF]Lemma,p.101] and a lemma which appears in his joint work with Bayer [BM], one has for any graded ideal L a procedure to estimate the regularity of U/L by induction on the dimension.

If k is infinite (one can reduce to this case), choose a general linear form and set r := reg(U/L + ( )) and r := reg(U/(L + (l)) sat ). Then

(i) reg(U/L) ≤ r + dim k (H 0 m (U/L) r ) if dim(U/L) ≥ 1, with H 0 m (U/L) r = (L sat /L) r ⊂ (U/L) r . (ii) reg(U/L sat ) ≤ r + dim k (H 1 m (U/L) r ), if dim(U/L) ≥ 2, with dim k H 1 m (U/L) r = dim k H 0 (X, O X (r )) -dim k (U/L sat )
, where X := proj(U/L). Estimates are precise and easy to get whenever dim(U/L) ≤ 1. Item (i) gives an estimate of the form (2d) (n-1)! for the regularity of an ideal generated in degrees at most d in a polynomial ring in n variables. Introducing (implicitely) an invariant measuring both the regularity and the defect of saturation, Caviglia-Sbarra [CS] proved a refinement of this estimate, reducing the exponent of 2d to 2 n-1 or close to 2 dim(U/L) .

To use (ii), one notices that whenever X is reduced, then proj(U/L + ( )) is reduced as well, and if the X i 's for i = 1, . . . , s are the irreducible components of X, then

dim k H 0 (X, O X (µ)) ≤ i dim k H 0 (X, O X i (µ)) and dim k H 0 (X, O X i (µ)) ≤ µ + dimX i dimX i e(X i )
for µ ≥ 0. This last estimate is valid for any symbolic power of a prime ideal (i.e., for the unmixed part of the scheme defined by a power of a prime ideal). If L is a homogeneous radical ideal or, more generally, an intersection of symbolic powers of prime ideals and e is the sum of their multiplicities, it implies reg(L) ≤ (e + 1) (dim(U/L))! .

The existence of a bound in these terms traces back to work of Kleiman (see Section 3]).

(2) It follows from results of Lazarsfeld that, in characteristic zero, if an homogeneous equidimensionnal ideal L of height h is defined in degree at most d, then there exists another ideal I with same radical such that L and I coincide locally at primes p such that (U/L) p is regular and reg(U/I) ≤ h(d -1). See Chardin-D'Cruz [CD] for a more precise statement.

As a consequence reg(U/L sat ) ≤ h(d -1) if the projective scheme defined by L has at most isolated singularities.

(3) Assume X ⊆ P n is reduced and equidimensionnal of dimension d over an algebraically closed field and c j is the number of connected components of scheme X(j) := X ∩ H 1 ∩ • • • ∩ H j for j = 0, • • • , d := dimX and general hyperplanes H i . Then c 0 ≤ c 1 ≤ • • • ≤ c d = deg(X) and X(d) is not contained in any linear space of dimension < n -σ with σ := d-1 j=0 c j . Thus deg(X) ≥ n -σ. Also c d-1 is bounded above by the number s of irreducible components of X. If X is the union of irreducible components X i , we thus have

s i=1 deg(X i ) + d-1 j=0 (c j -1) ≥ codim(X)
It follows that if X is connected in codimension r, then codim(X) ≤ deg(X) + (r -1)(s -1) ≤ r deg(X) -r + 1.

Prime ideals from designer ideals

In this section, we apply the method in [MP] to the designer ideals constructed by Ullery in [Ul].

First we define notation related to maximal shifts. Let T = k[x 1 , . . . , x n ] and let M be a finitely generated T -module. Set t T i (M ) -i .

We state a version of a result of Ullery:

Theorem 6.1 ([Ul, Theorem 1.3]). Let T = k[x 1 , . . . , x n ] and let M be a finitely generated T -module generated in a single non-negative degree with strictly increasing sequence of maximal graded shifts t T 0 (M ), t T 1 (M ), . . . , t T r (M ) . Set a = t T 0 (M ) and fix a positive integer N such that the number of elements in a minimal homogeneous generating set of M is ≤ N +a-1 a . Then there exists an ideal J M in S = T [y 1 , . . . , y N ] such that

t S i (J M ) = t T i+1 (M ) if 0 ≤ i ≤ r -1 t T r (M ) + i -r + 1 if r ≤ i ≤ N + r -1.
In particular, if we pick M to be generated in degree a = 1, and N sufficiently large, then there exists an ideal J M generated by homogeneous quadrics with any strictly increasing sequence as an initial sequence of its maximal graded shifts.

Theorem 2. 5 .

 5 In the notation and under the assumptions of Theorem 2.4 we have e Euler (M ) = e Hilb (M ) e Euler (M ) = e Euler (M ) e Hilb (M ) = e Hilb (M ) p i=1 deg(w i ) .

Proof.

  Note that dim (S[t]) = dim (S[It]) = dim S[It, t 2 ] = n + 1 since they are domains of the same transcendence degree. Since S[It] ⊂ S[It, t 2 ] ⊂ S[t] we conclude 0 < e Hilb (S[It]) ≤ e Hilb (S[It, t 2 ]) ≤ e Hilb (S[t]) = 1 .

  Theorem 4.3. Let M be an ideal generated by m ≥ 1 forms of the same degree d ≥ 2 in R = k[X 1 , . . . , X n ], and R M ∼ = R[M t] be the Rees Algebra of M which is considered as a standard graded quotient of the polynomial ring R[Y 1 , . . . , Y m ]. Then,

  t T i (M ) = max{j | Tor T i (M, k) j = 0} = max{j | β T ij (M ) = 0}.Thus t T 0 (M ) is the maximal degree of an element in a minimal generating set of M and t T 1 (M ) is the maximal degree of a minimal first syzygy of M . The maximal shifts t T i (M ) are related to regularity by reg(M ) = max 0≤i≤projdim(M )

  Consider the standard graded polynomial ring S = k[x 1 , . . . , x n ] over a field k. Let I be a homogeneous ideal minimally generated by forms f 1 , . . . , f m of degrees a 1 , . . . , a m , where m ≥ 2. Set d = max i {a i }. Consider a new ideal I generated by the forms {x d-a i f i } of degree d in the polynomial ring S = S[x].

Acknowledgements. We are very grateful to David Eisenbud for useful discussions.

Caviglia is partially supported by NSA MSP grant H98230-15-1-0004. Peeva is partially supported by NSF grant DMS-1702125.

We now take the step-by-step homogenization of the defining prime ideal of the Reeslike algebra of J M to produce prime ideals over any field with generators in degree at most 6 and arbitrarily large degree of first syzygies: Theorem 6.2. Fix a positive integer s ≥ 9 and field k. There exists a non-degenerate prime ideal P in a polynomial ring R over k with t R 0 (P ) = 6 and t R 1 (P ) = s.

Proof. Take T = k[x 1 , x 2 , x 3 ] and M = Ext 3 S T /(x 1 , x 2 , x 3 ) s-3 , S (4 -s) (where (4 -s) denotes a shift of degrees). Then M is a Cohen-Macaulay module with Betti table of the form:

where " * " denotes a non-zero entry and "-" denotes a zero entry. Thus,

Now let P be the step-by-step homogenization of the defining prime ideal of the Reeslike algebra of J in a larger polynomial ring R, as constructed in [MP]. By [START_REF] Mccullough | Counterexamples to the Eisenbud-Goto Regularity Conjecture[END_REF]Theorem 1.6],

t R 0 (P ) = max 2 t S 0 (J) + 1 , t S 1 (J) + 1 = max{6, 4} = 6. The structure of the minimal free resolution of P in [START_REF] Mccullough | Counterexamples to the Eisenbud-Goto Regularity Conjecture[END_REF]Theorem 3.10] implies that t R 1 (P ) = max 3 t S 0 (J) + 1 , t S 1 (J) + t S 0 (J) + 2, t S 2 (J) + 1 = max{9, 7, s} = s .

Ideals with Large Regularity

In this section we provide an infinite family of counterexamples to the Eisenbud-Goto Regularity Conjecture that do not rely on the Mayr-Meyer construction.

Proposition 7.1. Let T be a polynomial ring over a field k and let J = (f, g, h) be a homogeneous ideal of T such that f, g, h all have the same degree. Let x, y be new variables and set

Then reg S (S/I) ≥ reg T (T /J) + 4 .

Proof. Note that S/I is finitely generated as a T -module. In fact, x 2 y 2 (S/I) ∼ = (T /J)(-4) is a T -direct summand of S/I. Since S is faithfully flat over T , we have reg S (S/I) ≥ reg T (x 2 y 2 (S/I)) = reg T (T /J) + 4.

In [START_REF] Caviglia | Koszul Algebras, Castelnuovo-Mumford Regularity, and Generic initial ideals[END_REF]Example 4.2.1] Caviglia showed that if T = k[z 1 , z 2 , z 3 , z 4 ] and

We set S = T [x, y] and

By the previous proposition we see that reg(S/I) ≥ d 2 + 2 for d ≥ 2, while the degrees of the three generators of I are 3, 3, and d + 2. By [START_REF] Mccullough | Counterexamples to the Eisenbud-Goto Regularity Conjecture[END_REF]Theorem 1.6] we obtain the following result:

Theorem 7.2. Let P be the step-by-step homogenization of the Rees-like algebra of the ideal I above, in the polynomial ring R (as constructed in [MP]). Then In particular, the Eisenbud-Goto conjecture fails when d ≥ 34.