Resolution of Veronese Embedding of plane curves

TL;DR

This paper explicitly computes the minimal free resolution of the ideal of smooth plane curves of degree d under the Veronese embedding into P^5, providing detailed algebraic descriptions for degrees d ≥ 2.

Contribution

It provides the first explicit minimal free resolutions of the Veronese embeddings of smooth plane curves for all degrees d ≥ 2.

Findings

Explicit minimal free resolutions for d ≥ 2

Detailed algebraic structure of the ideals

Enhanced understanding of Veronese embeddings

Abstract

Let $C$ be a smooth (irreducible) curve of degree $d$ in $\mathbb{P}^{2}$. Let $\mathbb{P}^{2} \hookrightarrow \mathbb{P}^{5}$ be the Veronese embedding and let $\mathcal{I}_{C}$ denote the homogeneous ideal of $C$ on $\mathbb{P}^{5}$. In this note we explicitly write down the minimal free resolution of $\mathcal{I}_{C}$ for $d\geq 2