On Fano Schemes of Toric Varieties

Abstract

Abstract. Given a finite set of lattice points A, we consider the associated homogeneous binomial ideal IA and projective toric variety XA. We give a concise combinatorial description of all linear subspaces contained in the variety XA, or, equivalently, all solutions in linear forms to the system of binomial equations determined by IA. More precisely, we study the Fano scheme F k (XA) whose closed points correspond to k-dimensional linear spaces contained in XA. We show that the irreducible components of F k (XA) are in bijection to maximal Cayley structures for A of length at least k. We explicitly describe these irreducible components and their intersection behavior, characterize when F k (XA) is connected, and prove that if XA is smooth in dimension k, then every component of F k (XA) is smooth in its reduced structure. Furthermore, in the special case k = dim XA − 1, we describe the nonreduced structure of F k (XA).

Fano schemes of complete intersections in toric varieties

“…Let A ⊂ Z m be a finite collection of lattice points and Y = Y A ⊂ P n be the toric variety parametrized by the monomials corresponding to elements of A; here n = #A − 1. The main result of [22] states that irreducible components Z π,k of F k (Y A ) are in bijection with maximal Cayley structures π of length at least k, see Sect. 2.1.…”

“…with equality if ( † †) holds. For an arbitrary k-dimensional linear space L with [L] ∈ Z π,k , L is obtained from a subspace of some L π as above after acting by T , so the same dimension estimate for [22,Proposition 6.1]. We have seen above that every fiber of p 2 has dimension at least…”

“…Definition 2.1 (Definition 3.1 of [22]) A Cayley structure of length on A is a surjective map π : τ → preserving affine relations, where τ ≺ A and is the set of standard basis vectors e 0 , . .…”

“…Proof There is a face σ of τ for which π |σ is bijective; this follows from e.g. [22,Proposition 4.3]. For any e i ∈ , let v i ∈ σ be the unique element with π(v i ) = e i .…”

“…The Fano scheme for complete intersections which are general with respect to the property of containing a fixed linear space can be used to obtain identifiability results in machine learning [24]. Finally, the study of linear subspaces of projective toric varieties leads to a better understanding of A-discriminants [13,22].…”