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Computing graded Betti tables of toric surfaces. (English) Zbl 1476.14085

Summary: 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 40009. This allows us to formulate precise conjectures predicting what certain entries look like in the case of an arbitrary Veronese surface \(\nu _d(\mathbb{P}^2)\).

MSC:

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
13D02 Syzygies, resolutions, complexes and commutative rings
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)

Software:

Magma; SageMath
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Full Text: DOI arXiv

References:

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