Fano Schemes of Complete Intersections in Toric Varieties

TL;DR

This paper investigates the structure and dimension of Fano schemes of complete intersections within toric varieties, providing conditions for their non-emptiness, smoothness, and methods to count linear subspaces.

Contribution

It introduces a decomposition approach for Fano schemes in toric varieties and establishes criteria for their non-emptiness and smoothness, extending intersection theory techniques.

Findings

Decomposition of Fano schemes into closed subschemes based on irreducible components.

Conditions under which these subschemes are non-empty and smooth.

Method to count linear subspaces when the expected dimension is zero.

Abstract

We study Fano schemes $F_k(X)$ for complete intersections $X$ in a projective toric variety $Y\subset \mathbb{P}^n$. Our strategy is to decompose $F_k(X)$ into closed subschemes based on the irreducible decomposition of $F_k(Y)$ as studied by Ilten and Zotine. We define the expected dimension for these subschemes, which always gives a lower bound on the actual dimension. Under additional assumptions, we show that these subschemes are non-empty and smooth of the expected dimension. Using tools from intersection theory, we can apply these results to count the number of linear subspaces in $X$ when the expected dimension of $F_k(X)$ is zero.