Geometric Oversight

Zaqueu Ramos and Aron Simis
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Zaqueu Ramos: Federal University of Sergipe, Mathematics
Aron Simis: Federal University of Pernambuco, Mathematics

Chapter Chapter 3 in Determinantal Ideals of Square Linear Matrices, 2024, pp 53-73 from Springer

Abstract: Abstract In this chapter we give an overview of the main geometric players related to a projective hypersurface, among them the polar map and image, the gradient ideal and the Hessian matrix and its determinant. A discussion is enticed about the rank of the Jacobian matrix of a set of forms in a polynomial ring over a field, and a characteristic-free proof is given of the rank of the Grassmann Jacobian (i.e., the Jacobian matrix of the maximal minors of a generic matrix over a field). Complete coverage delivers the main properties of the determinant of the generic square matrix and the generic symmetric matrix over a field of characteristic ≠ 2 $$\neq 2$$ . Some consideration is given to the question as to when a projective hypersurface is defined by the determinant of a matrix of linear entries and how the algebraic features of this matrix as the ones in the book may reflect back into nontrivial traits of the hypersurface.

Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-55284-7_3

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DOI: 10.1007/978-3-031-55284-7_3

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