Computing graded Betti tables of toric surfaces
| Autor: | Jeroen Demeyer, Filip Cools, Wouter Castryck, Alexander Lemmens |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Surface (mathematics)
Pure mathematics Conjecture Mathematics::Commutative Algebra Applied Mathematics General Mathematics 010102 general mathematics Dimension (graph theory) Lattice (discrete subgroup) 01 natural sciences Upper and lower bounds Cohomology Mathematics - Algebraic Geometry Polygon FOS: Mathematics Mathematics - Combinatorics Combinatorics (math.CO) 0101 mathematics Algebraic Geometry (math.AG) Veronese surface Mathematics |
| Zdroj: | Transactions of the American Mathematical Society. 372:6869-6903 |
| ISSN: | 1088-6850 0002-9947 |
| DOI: | 10.1090/tran/7643 |
| Popis: | 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 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 ν 6 ( P 2 ) ⊆ P 27 \nu _6(\mathbb {P}^2) \subseteq \mathbb {P}^{27} in characteristic 40 009 40\,009 . This allows us to formulate precise conjectures predicting what certain entries look like in the case of an arbitrary Veronese surface ν d ( P 2 ) \nu _d(\mathbb {P}^2) . |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|