Secant Degree of Toric Surfaces and Delightful Planar Toric Degenerations

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.