Betti numbers of powers of ideals

Let A=K[x1, ..... ,xn] be a standard graded polynomial ring over a field K, let M = (x_1, .... , x_n) be the graded maximal ideal and I a graded ideal of A. For each i the Betti numbers b_i(I^k ) of I^k are polynomial functions for k>>0. We show that if I is M-primary, then these polynomial functions have the same degree for all i

Rational Surfaces with Anticanonical Divisor not Reduced

AbstractWe prove the finite generation of the monoid of effective divisor classes on a smooth projective rational surface X endowed with an anticanonical divisor such that all its irreducible components are of multiplicity one except one which has multiplicity two. In almost all cases, the self-intersection of a canonical divisor KX on X is strictly negative, hence - KX is neither ample nor numerically effective. In particular, X is not a Del Pezzo surface. Furthermore, it is shown that the first cohomology group of a numerically effective divisor vanishes; as a consequence, we determine the dimension of the complete linear system associated to any given divisor on

Ideals with linear quotients in Segre products

We establish that the Segre product between a polynomial ring on a field $K$ in $m$ variables and the second squarefree Veronese subalgebra of a polynomial ring on $K$ in $n$ variables has the intersection degree equal to three. We describe a class of monomial ideals of the Segre product with linear quotients

On sequences of integers for Hankel planes Σ of P^m

For a vector space R ⊆ k^{m+1}  of dimension r + 1 on the algebraically closed field  k we determine, for any i ≤ r, the possible numbers of Hankel i−planes contained in the r−plane P(R), linear space in P^m

s-Sequences and Monomial Modules

In this paper we study a monomial module M generated by an s-sequence and the main algebraic and homological invariants of the symmetric algebra of M. We show that the first syzygy module of a finitely generated module M, over any commutative Noetherian ring with unit, has a specific initial module with respect to an admissible order, provided M is generated by an s-sequence. Significant examples complement the results