Castenuovo bounds for certain subvarieties in \({\mathbb{P}}^ n\)
Let A be a finitely generated graded ring over an algebraically closed field k, generated by degree 1 elements. Let m denote the irrelevant maximal ideal and M a finitely generated graded A-module. The property of Buchsbaum is close to Cohen-Macaulay, but not quite. If \(\dim (M)=d\), M is Cohen-Macaulay if and only if \(H^ i_ m(M)=0\), \(i\neq d\); \(H^ i_ m(M)\) is the i-th local cohomology. Instead of giving a definition of Buchsbaum module, let me just point out that if M is Buchsbaum, then \(H^ i_ m(M)\) is annihilated by m, for \(i\neq d.\) The next concept needed is Castelnuovo regularity. [The geometrically inclined reader may look at \textit{D. Mumford}'s Princeton lecture notes, ``Curves on an algebraic surface'', to see the usefulness of the notion.] In local cohomology terms, M is r-regular if \([H^ i_ m(M)]^ j=0\), for \(i+j>m\) where the subscript denotes the j-th graded component. The authors prove the following: Let \(A=R/I\), where R is a polynomial ring in \(n+1\) variables and I a homogeneous ideal, A a Buchsbaum ring. Then: (i) regularity of A \(\leq\) degree of A. - (ii) If depth\((A)\geq 1\) then reg\((A)\leq \deg (A)-1\). - (iii) Let k be in addition of zero characteristic, and \(\dim(A)=d\geq 2\), A a domain. Let t be the smallest integer such that, \(\deg(A)-1\leq t\cdot(\text{rank}[A]_ 1-d)\). then reg\((A)\leq t.\) This gives a partial solution to the conjecture of Eisenbud and Goto. The authors also prove a proposition which asserts certain rings are Buchsbaum. This has the following corollary: ``Every reduced, irreducible and non degenerate curve \(X\) of degree \(\leq n+1\) in \({\mathbb{P}}^ n\) is arithmetically Buchsbaum, or Cohen-Macaulay''. Of course, the only non-trivial case is when \(\deg(X)=n+1\).
scientific article; zbMATH DE number 4021344
scientific article