On syzygies of divisors on rational normal scrolls
Let \(R\) denote the homogeneous polynomial ring of projective space \(\mathbb P^r\) over an algebraically closed field \(K\). Let \(Y \subset \mathbb P^r\) be a variety of minimal degree, i.e. one for which \(\deg(Y) = \text{codim}(Y)+1\). Such varieties have been completely classified. Let \(X \subset Y\) be a divisor, thought of as a subvariety of \(\mathbb P^n\), and assume that \(X\) is non-degenerate. Let \(I_X\) be the homogenous ideal of \(X\) in \(R\). The main object of study in this paper is the minimal free resolution of \(I_X\). It is known that if \(X\) is arithmetically Cohen-Macaulay (ACM) then the graded Betti numbers of \(I_X\) depend only on the degree of \(X\). This paper describes the graded Betti numbers in the non-ACM case, which is much more delicate. The graded Betti numbers are invariant in the divisor class of \(X\), and the author shows that the converse is almost true (giving the only exception). He carefully computes different important invariants in terms of the divisor class.
scientific article; zbMATH DE number 6353328