arXiv 1012.33962010-12-15math.AGmath.AC

一般平面曲线的行列式表示及其子格式

Determinantal representation and subschemes of general plane curves

Luca Chiantini, Juan Migliore

备注：: 14 pages

摘要：

令$M = (m_{ij})$是整数的$n \times n$方阵。出于我们的目的，我们可以不失一般性地假设$M$是齐次的，并且条目是非递增的。令$d$为任一对角线上的条目之和。我们给出了一个完整的表征，这样的矩阵具有的属性，即一般形式的程度$d$在$\mathbb C[x_0,x_1,x_2]$可以被写为行列式的矩阵形式$(f_{ij})$与$°f_{ij} = m_{ij}$（当然$f_{ij} = 0$，如果$m_{ij} < 0$）。作为结果，我们回答了与之相关的问题，即整数的$(n-1) \times n$矩阵$Q$具有这样的性质，即一个次数为$d$的一般平面曲线包含一个次数为Hilbert-Burch矩阵$Q$的零维子模式.这就引出了一种判定一般平面曲线所含线性级数性质的算法方法。

英文摘要：

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 $°f_{ij} = m_{ij}$ (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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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