Syzygies of Quadratic Veronese Embedding

TL;DR

This paper explicitly computes the syzygies of the quadratic Veronese embedding as group representations and constructs sheaf resolutions within the derived category, advancing understanding of these algebraic structures.

Contribution

It provides explicit calculations of syzygies and sheaf resolutions for the quadratic Veronese embedding, a novel detailed representation-theoretic analysis.

Findings

Explicit syzygy descriptions as GL(V) representations

Constructed sheaf resolutions in derived categories

Enhanced understanding of algebraic and geometric properties

Abstract

In this work we explicitly calculate syzygies of quadratic Veronese embedding $\mathbb{P}(V)\subset\mathbb{P}(\operatorname{Sym}^2V)$ as representations of the group $\operatorname{GL}(V)$. Also resolutions of the sheaves $\mathscr{O}_{\mathbb{P}(V)}(i)$ are constructed in the category $D(\mathbb{P}(\operatorname{Sym}^2V))$.