On Fano Schemes of Toric Varieties

TL;DR

This paper characterizes the components and structure of Fano schemes of toric varieties, linking them to maximal Cayley structures, and explores their smoothness and intersection properties.

Contribution

It provides explicit descriptions of Fano scheme components for toric varieties, characterizes their connectedness, and analyzes their smoothness and non-reduced structures.

Findings

Irreducible components correspond to maximal Cayley structures.

Characterization of when Fano schemes are connected.

Description of non-reduced structures for certain cases.

Abstract

Let $X_\mathcal{A}$ be the projective toric variety corresponding to a finite set of lattice points $\mathcal{A}$. We show that irreducible components of the Fano scheme $\mathbf{F}_k(X_\mathcal{A})$ parametrizing $k$-dimensional linear subspaces of $X_\mathcal{A}$ are in bijection to so-called maximal Cayley structures for $\mathcal{A}$. We explicitly describe these irreducible components and their intersection behaviour, characterize when $\mathbf{F}_k(X_\mathcal{A})$ is connected, and prove that if $X_\mathcal{A}$ is smooth in dimension $k$, then every component of $\mathbf{F}_k(X_\mathcal{A})$ is smooth in its reduced structure. Furthermore, in the special case $k=\dim X_\mathcal{A}-1$, we describe the non-reduced structure of $\mathbf{F}_k(X_\mathcal{A})$. Our main result is closely related to concurrent work done independently by Furukawa and Ito.