Determinantal representation and subschemes of general plane curves

TL;DR

This paper characterizes when general plane curves of a given degree can be represented as determinants of matrices of forms with specified degrees, and explores related subscheme containment properties.

Contribution

It provides a complete characterization of matrices that allow determinant representations of general plane curves and introduces an algorithmic approach to study linear series in these curves.

Findings

Characterization of matrices for determinant representations of plane curves

Answering which subschemes are contained in general plane curves

Development of an algorithmic method for properties of linear series

Abstract

Let $M = (m_{ij})$ be an $n \times n$ square matrix of integers. For our purposes, we can assume without loss of generality that $M$ is homogeneous and that the entries are non-increasing going leftward and downward. Let $d$ be the sum of the entries on either diagonal. We give a complete characterization of which such matrices have the property that a general form of degree $d$ in $\mathbb C[x_0,x_1,x_2]$ can be written as the determinant of a matrix of forms $(f_{ij})$ with $\deg f_{ij} = m_{ij}$ (of course $f_{ij} = 0$ if $m_{ij} < 0$). As a consequence, we answer the related question of which $(n-1) \times n$ matrices $Q$ of integers have the property that a general plane curve of degree $d$ contains a zero-dimensional subscheme whose degree Hilbert-Burch matrix is $Q$. This leads to an algorithmic method to determine properties of linear series contained in general plane curves.