Secant varieties of toric varieties arising from simplicial complexes

Abstract

Motivated by the study of the secant variety of the Segre-Veronese variety we propose a general framework to analyze properties of the secant varieties of toric embeddings of affine spaces defined by simplicial complexes. We prove that every such secant is toric, which gives a way to use combinatorial tools to study singularities. We focus on the Segre-Veronese variety for which we completely classify their secants that give Gorenstein or Q-Gorenstein varieties. We conclude providing the explicit description of the singular locus.

Cohen–Macaulay and Gorenstein tangential varieties of the Segre–Veronese varieties

“…Our approach is to use methods from [6] where the authors mainly deal with the secant varieties of the Segre–Veronese varieties. More details and examples can be found in [6]. Let $$\mathbb {N}$$ denote the set of non‐negative integers.…”

“…In this paper we allow vertices to have repeated labels. In this case $$\lbrace 1,\ldots ,n\rbrace$$ always refers to the labelling set of $$\Delta$$ rather than its vertex set; see [6, Example 2.1] for an example.…”

“…Furthermore, $$\sigma _2(X)$$ is known to be normal [19, Theorem 2.2] and Cohen–Macaulay (which followed from [6, Proposition 3.5]). On the other hand, the tangential variety $$\tau (X)$$ is not always normal [9, Example 2.22].…”

“…This paper is organized as follows. The first half of Section 2 is mainly based on [6] where we recall the background results needed later. In particular, we define toric varieties and simplicial embeddings.…”

“…where the sum is taken over all subsimplices 𝜎 ′ of 𝜎 including the empty simplex, and the product is taken over vertices 𝑖 ∈ 𝜎 ⧵ 𝜎 ′ ; see[6, Example 2.4] for an example.Proof of Theorem 2.2. In the following we write 𝜎 = 𝑣 for 𝜎 = {𝑣}, and regard 𝑥 𝜎 and 𝑦 𝜎 as polynomials in 𝑡 𝑖 and 𝑢 𝑖 .…”