Let M = (m_{ij}) be an nxn 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 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)xn 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.

Chiantini, L., Migliore, J. (2012). Determinantal representation and subschemes of general plane curves. LINEAR ALGEBRA AND ITS APPLICATIONS, 436, 1001-1013 [10.1016/j.laa.2011.06.011].

Determinantal representation and subschemes of general plane curves

CHIANTINI, LUCA;
2012-01-01

Abstract

Let M = (m_{ij}) be an nxn 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 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)xn 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.
Chiantini, L., Migliore, J. (2012). Determinantal representation and subschemes of general plane curves. LINEAR ALGEBRA AND ITS APPLICATIONS, 436, 1001-1013 [10.1016/j.laa.2011.06.011].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11365/23615
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