On Fano Schemes of Toric Varieties

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.

20 pages, 3 figures

References in corpus (4)

• Product Ranks of the $3\times 3$ Determinant and Permanent
• Fano schemes of determinants and permanents
• A combinatorial description of dual defects of toric varieties
• Fano Schemes of Lines on Toric Surfaces