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Problems and Results on Hilbert Functions of Graded Algebras

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Six Lectures on Commutative Algebra

Part of the book series: Progress in Mathematics ((MBC))

Abstract

Since a projective variety V = Z(I) ⊆ P n is an intersection of hypersurfaces, one of the most basic problems we can pose in relation to V is to describe the hypersurfaces containing it. In particular, one would like to know the maximal number of linearly independent hypersurfaces of each degree containing V, that is to know the dimension of I d , the vector space of homogeneous polynomials of degree d vanishing on V for various d. Since one knows the dimension \(\left( {\frac{{n + d}}{d}} \right)\) of the space of all forms of degree d, knowing the dimension of I d is equivalent to knowing the Hilbert function of the homogeneous coordinate ring A = k[X 0,..., X n ]/I of V, which is the vector space dimension of the degree d part of A.

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Authors and Affiliations

  1. Department of Mathematics, University of Genoa, Via Dodecaneso 35, 16146, Genoa, Italy

    Giuseppe Valla

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  1. Giuseppe Valla
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  1. Dept. Algebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, E-08007, Barcelona, Spain

    J. Elias , J. M. Giral , R. M. Miró-Roig  & S. Zarzuela , ,  & 

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Valla, G. (1998). Problems and Results on Hilbert Functions of Graded Algebras. In: Elias, J., Giral, J.M., Miró-Roig, R.M., Zarzuela, S. (eds) Six Lectures on Commutative Algebra. Progress in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0346-0329-4_5

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  • DOI: https://doi.org/10.1007/978-3-0346-0329-4_5

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0346-0328-7

  • Online ISBN: 978-3-0346-0329-4

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