Tate Resolutions for Segre Embeddings

TL;DR

This paper provides an explicit description of the Tate resolution for sheaves from Segre embeddings, revealing the nature of the maps involved, which originate from Sylvester and Bezout-type maps linked to the toric Jacobian.

Contribution

It offers a detailed, explicit construction of the Tate resolution for Segre embeddings, connecting the differentials to classical algebraic maps.

Findings

Explicit formulas for the Tate resolution terms and differentials.

Identification of the maps as Sylvester-type or Bezout-type.

Connection to the toric Jacobian in the resolution structure.

Abstract

We give an explicit description of the terms and differentials of the Tate resolution of sheaves arising from Segre embeddings of $\P^a\times\P^b$. We prove that the maps in this Tate resolution are either coming from Sylvester-type maps, or from Bezout-type maps arising from the so-called toric Jacobian.