Secant Degree of Toric Surfaces and Delightful Planar Toric Degenerations

Details Secant Degree of Toric Surfaces and Delightful Planar Toric Degenerations Secant Degree of Toric Surfaces and Delightful Planar Toric Degenerations

2010-12-11

arxiv.org/abs/1012.2454v1

Mathematics

Algebraic Geometry

19 pages, several figures

Scientific paper

The $k$-secant degree is studied with a combinatorial approach. A planar toric degeneration of any projective toric surface $X$ corresponds to a regular unimodular triangulation $D$ of the polytope defining $X$. If the secant ideal of the initial ideal with respect to $D$ coincides with the initial ideal of the secant ideal, then $D$ is said to be delightful and the $k$-secant degree of $X$ can be easily computed. All delightful triangulations of toric surfaces having sectional genus $g\leq1$ are completely classified and, for $g\geq2$, a lower bound for the $2$- and $3$-secant degree, by means of the combinatorial geometry and the singularities of non-delightful triangulations, is established.

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