Syzygies of general projections of canonical and paracanonical curves

TL;DR

This paper investigates the syzygies of projected canonical and paracanonical curves, focusing on their algebraic properties and whether they are defined by quadrics, supported by computational experiments.

Contribution

It introduces new results on the $ ilde{N}_{d,p}$ properties of projections of canonical and paracanonical curves, especially regarding their quadrics-cut nature.

Findings

Proposes conjectures on syzygy properties based on computational tests.

Analyzes the vanishing properties of Betti diagrams for projected curves.

Provides insights into the geometric information contained in syzygies of projected varieties.

Abstract

Let $X\subset\mathbb{P}^r$ be an integral linearly normal variety and $R=k[x_0,\cdots,x_r]$ the coordinate ring of $\mathbb{P}^r$. It is known that the syzygies of $X$ contain some geometric information. In recent years the syzygies of non-projectively normal varieties or in other words, the projection $X'$ of $X$ away from a linear subspace $W\subset\mathbb{P}^r$, were taken into considerations. Assuming that the coordinate ring of the ambient space that $X'$ lives in is $S$, there are two types of vanishing properties of the Betti diagrams of the projected varieties, the so-called $N_{d,p}^S$ and $\widetilde{N}_{d,p}$. The former one have been widely discussed for general varieties, for example by S. Kwak, Y. Choi and E. Park, while the latter one was discussed by W. Lee and E. Park for curves of very large degree. In this paper I will discuss about the $\widetilde{N}_{d,p}$ properties of the projection of a generic canonical and paracanonical curve away from a generic point and in particular whether they are cut out by quadrics. Some conjectures will be claimed based on the tests on Macaulay2.