Let X be a closed subscheme of the n-dimensional projective space Pn and let HF(X,-) and hp(X,-) denote, respectively, the Hilbert function and the Hilbert polynomial of X. We say that X has bipolynomial Hilbert function if HF(X,d) =min {hp(Pn, d), hp(X, d)} for every natural number d. We show that if X consists of a plane and generic lines, then X has bipolynomial Hilbert function. We also conjecture that generic configurations of non-intersecting linear spaces have bipolynomial Hilbert function.
Bipolynomial Hilbert functions
CATALISANO, MARIA VIRGINIA;GERAMITA, ANTHONY VITO
2010-01-01
Abstract
Let X be a closed subscheme of the n-dimensional projective space Pn and let HF(X,-) and hp(X,-) denote, respectively, the Hilbert function and the Hilbert polynomial of X. We say that X has bipolynomial Hilbert function if HF(X,d) =min {hp(Pn, d), hp(X, d)} for every natural number d. We show that if X consists of a plane and generic lines, then X has bipolynomial Hilbert function. We also conjecture that generic configurations of non-intersecting linear spaces have bipolynomial Hilbert function.
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