We study the Betti numbers of graded ideals containing the squares of the variables, in a polynomial ring. We prove the lex-plus-powers conjecture for such ideals.

Several open problems related to the behavior of the monoid of effective divisors and the nef cone for smooth projective surfaces over an algebraically closed field are discussed, motivating and putting into historical context concepts such as Mori dream spaces, Seshadri constants and the resurgence of homogeneous ideals in polynomial rings. Some recent work on these topics is discussed along w...

Star configurations are certain unions of linear subspaces of projective space. They have appeared in several different contexts: the study of extremal Hilbert functions for fat point schemes in the plane; the study of secant varieties of some classical algebraic varieties; the study of the resurgence of projective schemes. In this paper we study some algebraic properties of the ideals defining...