zbMATH — the first resource for mathematics

Max Noether’s theorem for integral curves. (English) Zbl 1386.14112
Authors’ abstract: We study a celebrated result of Max Noether on global sections of the \(n\)-dualizing sheaf of a smooth nonhyperelliptic curve in the case where the curve is integral. We reduce the proof of the statement in such a case to a purely numerical condition, which we show that holds if the non-Gorenstein points are bibranch at worst. This is our main result. We also extend the notion of a canonical embedding for integral curves with unibranch non-Gorenstein points at worst, in a way that we can express the dimensions of the components of the ideal in terms of the main invariants of the curve as well. Afterwards we focus on gonality, Clifford index and Koszul cohomology of non-Gorenstein curves by allowing torsion free sheaves of rank 1 in their definitions. We find an upper bound for the gonality, which agrees with Brill-Noether’s one for a rational and unibranch curve. We characterize curves of genus 5 with Clifford index 1, and, finally, we study Green’s conjecture for a certain class of curves, called nearly Gorenstein. The nearly Gorenstein part seems to be very interesting, but also the introduction is nice and useful.

14H20 Singularities of curves, local rings
14H45 Special algebraic curves and curves of low genus
14H51 Special divisors on curves (gonality, Brill-Noether theory)
Full Text: DOI
[1] Aprodu, M.; Farkas, G., Koszul cohomology and applications to moduli, Clay Math. Proc., 14, 25-50, (2011) · Zbl 1248.14039
[2] Arbarello, E.; Cornalba, M.; Griffiths, P. A.; Harris, J., Geometry of algebraic curves, (1985), Springer-Verlag · Zbl 0559.14017
[3] Arbarello, E.; Sernesi, E., Petri’s approach to the study of the ideal associated to a special divisor, Invent. Math., 49, 99-119, (1978) · Zbl 0399.14019
[4] Babbage, D. W., A note on the quadrics through a canonical curve, J. Lond. Math. Soc., 14, 310-315, (1939) · JFM 65.1398.03
[5] Ballico, E., Bril-Noether theory for rank 1 torsion free sheaves on singular projective curves, J. Korean Math. Soc., 37, 359-369, (2000) · Zbl 0993.14016
[6] Ballico, E.; Fontanari, C.; Tasin, L., Koszul cohomology and singular curves, (29 Sep 2009)
[7] Barucci, V.; D’Anna, M.; Fröberg, R., Analytically unramified one-dimensional semilocal rings and their value semigroups, J. Pure Appl. Algebra, 147, 215-254, (2000) · Zbl 0963.13021
[8] Barucci, V.; Fröberg, R., One-dimensional almost Gorenstein rings, J. Algebra, 88, 418-442, (1997) · Zbl 0874.13018
[9] Catanese, F., Pluricanonical-Gorenstein-curves, (Enumerative Geometry and Classical Algebraic Geometry, Nice, 1981, Progr. Math., vol. 24, (1982)), 51-95
[10] Contiero, A.; Stoehr, K.-O., Upper bounds for the dimension of moduli spaces of curves with symmetric Weierstrass semigroups, J. Lond. Math. Soc., 88, 580-598, (2013) · Zbl 1288.14016
[11] Deligne, P.; Mumford, D., The irreducibility of the space of curves of a given genus, Publ. Math. Inst. Hautes Études Sci., 36, 75-109, (1969) · Zbl 0181.48803
[12] Coppens, M., Free linear systems on integral Gorenstein curves, J. Algebra, 145, 209-218, (1992) · Zbl 0770.14002
[13] Eisenbud, D., The geometry of syzygies, (2005), Springer-Verlag
[14] Eisenbud, D.; Harris, J.; Koh, J.; Stillman, M., Determinantal equations for curves of high degree, Amer. J. Math., 110, 513-539, (1988) · Zbl 0681.14027
[15] Enriques, F., Sulle curve canoniche di genera p cello spazio a \(p - 1\) dimensioni, Rend. Accad. Sci. Ist. Bologna, 23, 80-82, (1919)
[16] Feital, L., Gonalidade e o teorema de MAX Noether para curvas não-Gorenstein, Ph.D. Thesis
[17] Feital, L.; Martins, R. V., Gonality of non-Gorenstein curves of genus five, Bull. Braz. Math. Soc., 45, 4, 1-22, (2014)
[18] Franciosi, M.; Tenni, E., Green’s conjecture for binary curves
[19] Fujita, T., On hyperelliptic polarized varieties, Tohoku Math. J., 35, 1-44, (1983) · Zbl 0494.14003
[20] Green, M., Koszul cohomology and the geometry of projective varieties, J. Differential Geom., 19, 125-171, (1984) · Zbl 0559.14008
[21] Hartshorne, R., Generalized divisors on Gorenstein curves and a theorem of Noether, J. Math. Kyoto Univ., 26, 3, 375-386, (1986) · Zbl 0613.14008
[22] Kleiman, S. L.; Martins, R. V., The canonical model of a singular curve, Geom. Dedicata, 139, 139-166, (2009) · Zbl 1172.14019
[23] Martins, R. V., A generalization of MAX Noether’s theorem, Proc. Amer. Math. Soc., 140, 377-391, (2012) · Zbl 1234.14025
[24] Martins, R. V., Trigonal non-Gorenstein curves, J. Pure Appl. Algebra, 209, 873-882, (2007) · Zbl 1108.14026
[25] Mumford, D.; Saint-Donat, B., Toroidal embeddings I, Lecture Notes in Math., vol. 339, 209, (1973), Springer-Verlag Berlin-Heidelberg-New York, VIII · Zbl 0271.14017
[26] Mumford, D., Curves and their Jacobians, (1975), The University of Michigan Press Ann Arbor · Zbl 0316.14010
[27] Noether, M., Über die invariante darstellung algebraicher funktionen, Math. Ann., 17, 263-284, (1880)
[28] Petri, K., Über die invariante darstellung algebraischer funktionen eiener veränderlichen, Math. Ann., 88, 242-289, (1922) · JFM 49.0264.02
[29] Rosa, R.; Stoehr, K-O., Trigonal Gorenstein curves, J. Pure Appl. Algebra, 174, 187-205, (2002) · Zbl 1059.14038
[30] Rosenlicht, M., Equivalence relations on algebraic curves, Ann. of Math., 56, 169-191, (1952) · Zbl 0047.14503
[31] Saint-Donat, B., On Petri’s analysis of the linear system of quadrics through a canonical curve, Math. Ann., 206, 157-175, (1973) · Zbl 0315.14010
[32] Saki, F., Canonical models of complements of stable curves, (Proc. Int. Symp. on Algebraic Geometry, Kyoto, (1977)), 643-661
[33] Shokourov, V. V., The Noether-Enriques theorem on canonical curves, Mat. Sb., 86, 367-408, (1972) · Zbl 0225.14017
[34] Stoehr, K.-O.; Viana, P., A variant of Petri’s analysis of the canonical ideal of an algebraic curve, Manuscripta Math., 61, 223-248, (1988) · Zbl 0661.14025
[35] Stoehr, K.-O., On the poles of regular differentials of singular curves, Bull. Braz. Math. Soc., 24, 105-135, (1993)
[36] Stoehr, K.-O., On the moduli spaces of Gorenstein curves with symmetric Weierstrass semigroups, J. Reine Angew. Math., 441, 189-213, (1993)
[37] Voisin, C., Green’s generic Syzygy conjecture for curves of even genus lying on a K3 surface, J. Eur. Math. Soc., 4, 363-404, (2002) · Zbl 1080.14525
[38] Voisin, C., Green’s canonical Syzygy conjecture for generic curves of odd genus, Compos. Math., 141, 1163-1190, (2005) · Zbl 1083.14038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.