Problems and Results on Hilbert Functions of Graded Algebras

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 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada % Wcaaqaaiaad6gacqGHRaWkcaWGKbaabaGaamizaaaaaiaawIcacaGL % Paaaaaa!3B34! $$\left( {\frac{{n + d}}{d}} \right)$$ 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.