Equations of some embeddings of a projective space into another one

TL;DR

This paper investigates equations defining embeddings of projective spaces, focusing on ideals generated by minors of Jacobian dual matrices, and explores their resolutions to address a conjecture on regularity.

Contribution

It proves that the ideal of maximal minors defines the image scheme and discusses the implications of its linear resolution for a conjecture on regularity.

Findings

The ideal of maximal minors defines the image scheme.

The ideal is generated in degree n.

Linearity of the ideal's resolution relates to the conjecture.

Abstract

In arXiv:math/0405373 , Eisenbud, Huneke and Ulrich conjectured a result on the Castelnuovo-Mumford regularity of the embedding of a projective space $\mathbb{P}^{n-1}\hookrightarrow \mathbb{P}^{r-1}$ determined by generators of a linearly presented $\mathfrak{m}$-primary ideal. This result implies in particular that the image is scheme defined by equations of degree at most $n$. In this text we prove that the ideal of maximal minors of the Jacobian dual matrix associated to the input ideal defines the image as a scheme; it is generated in degree $n$. Showing that this ideal has a linear resolution would imply that the conjecture in arXiv:math/0405373 holds. Furthermore, if this ideal of minors coincides with the one of the image in degree $n$ - what we hope to be true - the linearity of the resolution of this ideal of maximal minors is equivalent to the conjecture in arXiv:math/0405373.