## Computing graded Betti tables of toric surfaces

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- by Wouter Castryck, Filip Cools, Jeroen Demeyer and Alexander Lemmens PDF
- Trans. Amer. Math. Soc.
**372**(2019), 6869-6903 Request permission

## Abstract:

We present various facts on the graded Betti table of a projectively embedded toric surface, expressed in terms of the combinatorics of its defining lattice polygon. These facts include explicit formulas for a number of entries, as well as a lower bound on the length of the quadratic strand that we conjecture to be sharp (and prove to be so in several special cases). We also present an algorithm for determining the graded Betti table of a given toric surface by explicitly computing its Koszul cohomology and report on an implementation in SageMath. It works well for ambient projective spaces of dimension up to roughly $25$, depending on the concrete combinatorics, although the current implementation runs in finite characteristic only. As a main application we obtain the graded Betti table of the Veronese surface $\nu _6(\mathbb {P}^2) \subseteq \mathbb {P}^{27}$ in characteristic $40 009$. This allows us to formulate precise conjectures predicting what certain entries look like in the case of an arbitrary Veronese surface $\nu _d(\mathbb {P}^2)$.## References

- Marian Aprodu and Jan Nagel,
*Koszul cohomology and algebraic geometry*, University Lecture Series, vol. 52, American Mathematical Society, Providence, RI, 2010. MR**2573635**, DOI 10.1090/ulect/052 - Adam Boocher, Wouter Castryck, Milena Hering and Alexander Lemmens,
*Torus weights of resolutions of Veronese embeddings*, in preparation. - Wieb Bosma, John Cannon, and Catherine Playoust,
*The Magma algebra system. I. The user language*, J. Symbolic Comput.**24**(1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR**1484478**, DOI 10.1006/jsco.1996.0125 - Juliette Bruce, Daniel Erman, Steve Goldstein, and Jay Yang,
*Conjectures and computations of Veronese syzygies*, Experimental Mathematics, to appear. - Winfried Bruns, Aldo Conca, and Tim Römer,
*Koszul homology and syzygies of Veronese subalgebras*, Math. Ann.**351**(2011), no. 4, 761–779. MR**2854112**, DOI 10.1007/s00208-010-0616-1 - Winfried Bruns, Joseph Gubeladze, and Ngô Viêt Trung,
*Normal polytopes, triangulations, and Koszul algebras*, J. Reine Angew. Math.**485**(1997), 123–160. MR**1442191** - Wouter Castryck,
*Moving out the edges of a lattice polygon*, Discrete Comput. Geom.**47**(2012), no. 3, 496–518. MR**2891244**, DOI 10.1007/s00454-011-9376-2 - Wouter Castryck,
*A lower bound for the gonality conjecture*, Mathematika**63**(2017), no. 2, 561–563. MR**3706597**, DOI 10.1112/S0025579317000067 - Wouter Castryck and Filip Cools,
*Newton polygons and curve gonalities*, J. Algebraic Combin.**35**(2012), no. 3, 345–366. MR**2892979**, DOI 10.1007/s10801-011-0304-6 - Wouter Castryck and Filip Cools,
*A minimal set of generators for the canonical ideal of a nondegenerate curve*, J. Aust. Math. Soc.**98**(2015), no. 3, 311–323. MR**3337885**, DOI 10.1017/S1446788714000573 - Wouter Castryck and Filip Cools,
*Linear pencils encoded in the Newton polygon*, Int. Math. Res. Not. IMRN**10**(2017), 2998–3049. MR**3658131**, DOI 10.1093/imrn/rnw082 - Wouter Castryck, Filip Cools, Jeroen Demeyer, and Alexander Lemmens,
*Canonical syzygies of smooth curves on toric surfaces*, J. Pure Appl. Algebra**224**(2020), no. 2, 507–527. MR**3987963**, DOI 10.1016/j.jpaa.2019.05.018 - Filip Cools and Alexander Lemmens,
*Minimal polygons with fixed lattice width*, Ann. Comb.**23**(2019), no. 2, 285–293. MR**3962858**, DOI 10.1007/s00026-019-00431-0 - David A. Cox, John B. Little, and Henry K. Schenck,
*Toric varieties*, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR**2810322**, DOI 10.1090/gsm/124 - Lawrence Ein, Daniel Erman, and Robert Lazarsfeld,
*A quick proof of nonvanishing for asymptotic syzygies*, Algebr. Geom.**3**(2016), no. 2, 211–222. MR**3477954**, DOI 10.14231/AG-2016-010 - Lawrence Ein and Robert Lazarsfeld,
*Asymptotic syzygies of algebraic varieties*, Invent. Math.**190**(2012), no. 3, 603–646. MR**2995182**, DOI 10.1007/s00222-012-0384-5 - Lawrence Ein and Robert Lazarsfeld,
*The gonality conjecture on syzygies of algebraic curves of large degree*, Publ. Math. Inst. Hautes Études Sci.**122**(2015), 301–313. MR**3415069**, DOI 10.1007/s10240-015-0072-2 - David Eisenbud,
*The geometry of syzygies*, Graduate Texts in Mathematics, vol. 229, Springer-Verlag, New York, 2005. A second course in commutative algebra and algebraic geometry. MR**2103875** - David Eisenbud and Joe Harris,
*On varieties of minimal degree (a centennial account)*, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 3–13. MR**927946**, DOI 10.1090/pspum/046.1/927946 - Gavril Farkas and Michael Kemeny,
*Linear syzygies of curves with prescribed gonality*, preprint. - L. Fejes Tóth and E. Makai Jr.,
*On the thinnest non-separable lattice of convex plates*, Studia Sci. Math. Hungar.**9**(1974), 191–193 (1975). MR**370369** - William Fulton,
*Introduction to toric varieties*, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR**1234037**, DOI 10.1515/9781400882526 - Francisco Javier Gallego and B. P. Purnaprajna,
*Some results on rational surfaces and Fano varieties*, J. Reine Angew. Math.**538**(2001), 25–55. MR**1855753**, DOI 10.1515/crll.2001.068 - Ornella Greco and Ivan Martino,
*Syzygies of the Veronese modules*, Comm. Algebra**44**(2016), no. 9, 3890–3906. MR**3503390**, DOI 10.1080/00927872.2015.1027389 - Mark L. Green,
*Koszul cohomology and the geometry of projective varieties*, J. Differential Geom.**19**(1984), no. 1, 125–171. MR**739785** - Mark Green and Robert Lazarsfeld,
*On the projective normality of complete linear series on an algebraic curve*, Invent. Math.**83**(1986), no. 1, 73–90. MR**813583**, DOI 10.1007/BF01388754 - Christian Haase, Benjamin Nill, Andreas Paffenholz, and Francisco Santos,
*Lattice points in Minkowski sums*, Electron. J. Combin.**15**(2008), no. 1, Note 11, 5. MR**2398828** - Christian Haase and Josef Schicho,
*Lattice polygons and the number $2i+7$*, Amer. Math. Monthly**116**(2009), no. 2, 151–165. MR**2478059**, DOI 10.4169/193009709X469913 - Milena S. Hering,
*Syzygies of toric varieties*, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)–University of Michigan. MR**2708466** - Robert J. Koelman,
*The number of moduli of families of curves on toric surfaces*, Ph.D. thesis, Katholieke Universiteit Nijmegen, 1991. - Robert Jan Koelman,
*A criterion for the ideal of a projectively embedded toric surface to be generated by quadrics*, Beiträge Algebra Geom.**34**(1993), no. 1, 57–62. MR**1239278** - Sijong Kwak,
*Castelnuovo regularity for smooth subvarieties of dimensions $3$ and $4$*, J. Algebraic Geom.**7**(1998), no. 1, 195–206. MR**1620706** - Margherita Lelli-Chiesa,
*Green’s conjecture for curves on rational surfaces with an anticanonical pencil*, Math. Z.**275**(2013), no. 3-4, 899–910. MR**3127041**, DOI 10.1007/s00209-013-1164-7 - Alexander Lemmens,
*Combinatorial aspects of syzygies of toric varieties*, Ph.D. thesis, KU Leuven, 2018. - Alexander Lemmens,
*On the $n$-th row of the graded Betti table of an $n$-dimensional toric variety*, J. Algebraic Combin.**47**(2018), no. 4, 561–584. MR**3813640**, DOI 10.1007/s10801-017-0786-y - LinBox,
*Exact linear algebra over the integers and finite fields*, Version 1.3.2, http://linalg.org/ (2012). - Frank Loose,
*On the graded Betti numbers of plane algebraic curves*, Manuscripta Math.**64**(1989), no. 4, 503–514. MR**1005250**, DOI 10.1007/BF01170942 - Niels Lubbes and Josef Schicho,
*Lattice polygons and families of curves on rational surfaces*, J. Algebraic Combin.**34**(2011), no. 2, 213–236. MR**2811147**, DOI 10.1007/s10801-010-0268-y - Giorgio Ottaviani and Raffaella Paoletti,
*Syzygies of Veronese embeddings*, Compositio Math.**125**(2001), no. 1, 31–37. MR**1818055**, DOI 10.1023/A:1002662809474 - Euisung Park,
*On syzygies of Veronese embedding of arbitrary projective varieties*, J. Algebra**322**(2009), no. 1, 108–121. MR**2526378**, DOI 10.1016/j.jalgebra.2009.03.022 - Jürgen Rathmann,
*An effective bound for the gonality conjecture*, preprint. - Elena Rubei,
*A result on resolutions of Veronese embeddings*, Ann. Univ. Ferrara Sez. VII (N.S.)**50**(2004), 151–165 (English, with English and Italian summaries). MR**2159811** - SageMath, the Sage Mathematics Software System, Version 7.2, http://www.sagemath.org/ (2016).
- Hal Schenck,
*Lattice polygons and Green’s theorem*, Proc. Amer. Math. Soc.**132**(2004), no. 12, 3509–3512. MR**2084071**, DOI 10.1090/S0002-9939-04-07523-9 - David H. Yang,
*$S_n$-equivariant sheaves and Koszul cohomology*, Res. Math. Sci.**1**(2014), Art. 10, 6. MR**3375645**, DOI 10.1186/s40687-014-0010-9

## Additional Information

**Wouter Castryck**- Affiliation: Departement Wiskunde, KU Leuven, Celestijnenlaan 200B, Leuven, Belgium; Departement ESAT, KU Leuven and imec, Kasteelpark Arenberg 10/2452, Leuven, Belgium; Vakgroep Wiskunde, Universiteit Gent, Krijgslaan 281, Gent, Belgium
- MR Author ID: 800098
- Email: wouter.castryck@esat.kuleuven.be
**Filip Cools**- Affiliation: Departement Wiskunde, KU Leuven, Celestijnenlaan 200B, Leuven, Belgium
- MR Author ID: 771467
- Email: f.cools@kuleuven.be
**Jeroen Demeyer**- Affiliation: Vakgroep Wiskunde, Universiteit Gent, Krijgslaan 281, Gent, Belgium
- MR Author ID: 782465
- Email: jeroen.demeyer@ugent.be
**Alexander Lemmens**- Affiliation: Departement Wiskunde, KU Leuven, Celestijnenlaan 200B, Leuven, Belgium
- MR Author ID: 1273465
- Email: alexander.lemmens@kuleuven.be
- Received by editor(s): December 2, 2016
- Received by editor(s) in revised form: June 4, 2018, and June 13, 2018
- Published electronically: August 20, 2019
- Additional Notes: This research was partially supported by the research project G093913N of the Research Foundation Flanders (FWO), by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) with ERC Grant Agreement No. 615722 MOTMELSUM, and by the Labex CEMPI (ANR-11-LABX-0007-01)

The fourth author was supported by a Ph.D. fellowship of the Research Foundation Flanders (FWO) - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**372**(2019), 6869-6903 - MSC (2010): Primary 14M25, 13D02, 52B20
- DOI: https://doi.org/10.1090/tran/7643
- MathSciNet review: 4024541