Abstract
The Minimal Resolution Conjecture (MRC) for points on a projective variety X predicts that the Betti numbers of general sets of points in X are as small as the geometry (Hilbert function) of X allows. To a large extent, we settle this conjecture for a curve C with general moduli. We then proceed to find a full solution to the Ideal Generation Conjecture for curves with general moduli. In a different direction, we prove that
Funding statement: The first author was partly supported by the CNCS-UEFISCDI grant PN-II-PCE-2011-3-0288 and by a Humboldt fellowship. The second and third authors were partly supported by the SFB 647 “Raum-Zeit-Materie”.
Acknowledgements
The first author thanks the Max-Planck-Institut für Mathematik Bonn and the Humboldt-Universität zu Berlin for hospitality during the preparation of this work. The second author is grateful to Frank Schreyer for very useful discussions on matters related to this circle of ideas. We thank the referee for a number of pertinent comments that clearly improved the presentation of this paper.
References
[1]
M. Aprodu and G. Farkas,
The Green conjecture for smooth curves lying on arbitrary
[2] A. Beauville, Determinantal hypersurfaces, Michigan Math. J. 48 (2000), 39–64. 10.1307/mmj/1030132707Search in Google Scholar
[3] A. Beauville, Some stable vector bundles with reducible theta divisors, Manuscripta Math. 110 (2003), 343–349. 10.1007/s00229-002-0341-5Search in Google Scholar
[4] J. Brennan, J. Herzog and B. Ulrich, Maximally generated Cohen–Macaulay modules, Math. Scand. 61 (1987), 181–203. 10.7146/math.scand.a-12198Search in Google Scholar
[5] M. Casanellas and R. Hartshorne, ACM bundles on cubic surfaces, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 3, 709–731. 10.4171/JEMS/265Search in Google Scholar
[6] M. Casanellas and R. Hartshorne, Stable Ulrich bundles, Internat. J. Math. 23 (2012), Article ID 1250083. 10.1142/S0129167X12500838Search in Google Scholar
[7]
A. Chiodo, D. Eisenbud, G. Farkas and F.-O. Schreyer,
Syzygies of torsion bundles and the geometry of the level
[8] C. Ciliberto and G. Pareschi, Pencils of minimal degree on curves on a K3 surface, J. reine angew. Math. 460 (1995), 15–36. Search in Google Scholar
[9] M. Coppens and G. Martens, Linear series on general k-gonal curves, Abh. Math. Semin. Univ. Hambg. 69 (1999), 347–371. 10.1007/BF02940885Search in Google Scholar
[10] E. Coskun, Ulrich bundles on quartic surfaces with Picard number 1, C. R. Acad. Sci. Paris Sér. I 351 (2013), 221–224. 10.1016/j.crma.2013.04.005Search in Google Scholar
[11] E. Coskun and R. Kulkarni and Y. Mustopa, Pfaffian quartic surfaces and representations of Clifford algebras, Doc. Math. 17 (2012), 1003–1028. Search in Google Scholar
[12] E. Coskun, R. Kulkarni and Y. Mustopa, The geometry of Ulrich bundles on del Pezzo surfaces, J. Algebra 375 (2013), 280–301. 10.1016/j.jalgebra.2012.08.032Search in Google Scholar
[13] L. Ein and R. Lazarsfeld, Stability and restrictions of Picard bundles with an application to the normal bundles of elliptic curves, Complex projective geometry, London Math. Soc. Lecture Note Ser. 179, Cambridge University Press, Cambridge (1992), 149–156. 10.1017/CBO9780511662652.011Search in Google Scholar
[14] D. Eisenbud, G. Fløystad and F.-O. Schreyer, Sheaf cohomology and sheaf resolutions over exterior algebras, Trans. Amer. Math. Soc. 353 (2003), 4397–4426. 10.1090/S0002-9947-03-03291-4Search in Google Scholar
[15] D. Eisenbud, S. Popescu, F.-O. Schreyer and C. Walter, Exterior algebra methods for the minimal resolution conjecture, Duke Math. J. 112 (2002), 379–395. 10.1215/S0012-9074-02-11226-5Search in Google Scholar
[16] D. Eisenbud and F.-O. Schreyer, Resultants and Chow forms via exterior syzygies, J. Amer. Math. Soc. 16 (2003), 537–579. 10.1090/S0894-0347-03-00423-5Search in Google Scholar
[17] D. Eisenbud and F.-O. Schreyer, Boij–Söderberg theory, Combinatorial aspects of commutative algebra, Springer, Berlin (2011), 35–48. 10.1007/978-3-642-19492-4_3Search in Google Scholar
[18]
G. Farkas, M. Mustaţă and M. Popa,
Divisors on
[19] M. Green, Koszul cohomology and the cohomology of projective varieties, J. Differential Geom. 19 (1984), 125–171. 10.4310/jdg/1214438426Search in Google Scholar
[20]
M. Green and R. Lazarsfeld,
Special divisors on curves on a
[21]
A. Hirschowitz and C. Simpson,
La résolution minimale de l’arrangement d’un grand nombre de points dans
[22] R. Lazarsfeld, A sampling of vector bundle techniques in the study of linear series, Lectures on Riemann surfaces (Trieste 1987), World Scientific, Teaneck (1989), 500–559. 10.1142/9789814503365_0011Search in Google Scholar
[23] R. Lazarsfeld, Lectures on linear series, Complex algebraic geometry (Park City 1993), IAS/Park City Math. Ser. 3, American Mathematical Society, Providence (1997), 163–224. 10.1090/pcms/003/03Search in Google Scholar
[24] R. Miró-Roig and J. Pons-Llopis, The minimal resolution conjecture for points on del Pezzo surfaces, Algebra Number Theory 6 (2012), 27–46. 10.2140/ant.2012.6.27Search in Google Scholar
[25] S. Mukai, Biregular classification of Fano 3-folds and Fano manifolds of coindex 3, Proc. Natl. Acad. Sci. USA 86 (1989), 3000–3002. 10.1073/pnas.86.9.3000Search in Google Scholar
[26] M. Mustaţă, Graded Betti numbers of general finite subsets of points on projective varieties, Matematiche (Catania) 53 (1998), 53–81. Search in Google Scholar
[27] M. Popa, On the base locus of the generalized theta divisor, C. R. Acad. Sci. Paris Sér. I 329 (1999), 507–512. 10.1016/S0764-4442(00)80051-8Search in Google Scholar
[28] M. Raynaud, Sections des fibrés vectoriels sur une courbe, Bull. Soc. Math. France 110 (1982), 103–125. 10.24033/bsmf.1955Search in Google Scholar
[29] E. Sernesi, On the existence of certain families of curves, Invent. Math. 75 (1984), 25–57. 10.1007/BF01403088Search in Google Scholar
[30] C. Voisin, Green’s canonical syzygy conjecture for generic curves of odd genus, Compos. Math. 141 (2005), 1163–1190. 10.1112/S0010437X05001387Search in Google Scholar
© 2017 Walter de Gruyter GmbH, Berlin/Boston
