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Geometric Oversight

In: Determinantal Ideals of Square Linear Matrices

Author

Listed:
  • Zaqueu Ramos

    (Federal University of Sergipe, Mathematics)

  • Aron Simis

    (Federal University of Pernambuco, Mathematics)

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.

Suggested Citation

  • Zaqueu Ramos & Aron Simis, 2024. "Geometric Oversight," Springer Books, in: Determinantal Ideals of Square Linear Matrices, chapter 0, pages 53-73, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-55284-7_3
    DOI: 10.1007/978-3-031-55284-7_3
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