On base loci of higher fundamental forms of toric varieties

TL;DR

This paper investigates the base loci of higher fundamental forms of toric varieties, establishing criteria based on the geometric configuration of points and providing classifications and examples related to Mori dream spaces.

Contribution

It characterizes when the higher fundamental forms of toric varieties have non-empty base loci and applies this to classify second fundamental forms on toric surfaces and construct examples of non-Mori dream blowups.

Findings

Higher fundamental forms have non-empty base locus iff points lie on a degree-m affine hypersurface.

Classification of second fundamental forms on toric surfaces.

Examples of weighted 3-dimensional projective spaces with non-Mori dream blowups.

Abstract

We study the base locus of the higher fundamental forms of a projective toric variety $X$ at a general point. More precisely we consider the closure $X$ of the image of a map $({\mathbb C}^*)^k\to {\mathbb P}^n$, sending $t$ to the vector of Laurent monomials with exponents $p_0,\dots,p_n\in {\mathbb Z}^k$. We prove that the $m$-th fundamental form of such an $X$ at a general point has non empty base locus if and only if the points $p_i$ lie on a suitable degree-$m$ affine hypersurface. We then restrict to the case in which the points $p_i$ are all the lattice points of a lattice polytope and we give some applications of the above result. In particular we provide a classification for the second fundamental forms on toric surfaces, and we also give some new examples of weighted $3$-dimensional projective spaces whose blowing up at a general point is not Mori dream.