Toric varieties and Gröbner bases: the complete Q -factorial case
| Autor: | Lea Terracini, Michele Rossi |
|---|---|
| Přispěvatelé: | Rossi, M, Terracini, L |
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Factorial
Pure mathematics Algebra and Number Theory Conjecture Applied Mathematics Gale duality Gröbner fan Initial ideals Secondary fan Toric ideals Toric varieties 010102 general mathematics Projective embedding Toric varietie 010103 numerical & computational mathematics Initial ideal Commutative Algebra (math.AC) 01 natural sciences Toric ideal FOS: Mathematics 0101 mathematics Algebraic Geometry (math.AG) Mathematics |
| Popis: | We present two algorithms determining all the complete and simplicial fans admitting a fixed non-degenerate set of vectors $V$ as generators of their 1-skeleton. The interplay of the two algorithms allows us to discerning if the associated toric varieties admit a projective embedding, in principle for any values of dimension and Picard number. The first algorithm is slower than the second one, but it computes all complete and simplicial fans supported by $V$ and lead us to formulate a topological-combinatoric conjecture about the definition of a fan. On the other hand, we adapt the Sturmfels' arguments on the Gröbner fan of toric ideals to our complete case; we give a characterization of the Gröbner region and show an explicit correspondence between Gröbner cones and chambers of the secondary fan. A homogenization procedure of the toric ideal associated to $V$ allows us to employing GFAN and related software in producing our second algorithm. The latter turns out to be much faster than the former, although it can compute only the projective fans supported by $V$. We provide examples and a list of open problems. In particular we give examples of rationally parametrized families of $\Q$-factorial complete toric varieties behaving in opposite way with respect to the dimensional jump of the nef cone over a special fibre. 18 pages, 2 figures. Final version accepted for publication in the special issue of the Journal AAAECC, concerning "Algebraic Geometry from an Algorithmic point of View" |
| Databáze: | OpenAIRE |
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