Abstract. Motivated by Stillman's question, we show that the projective dimension of an ideal generated by four quadric forms in a polynomial ring is at most 6; moreover, this bound is tight. We achieve this bound, in part, by giving a characterization of the low degree generators of ideals primary to height three primes of multiplicities one and two.

ABSTRACT. We present a closed formula and a simple algorithmic procedure to compute the projective dimension of square-free monomial ideals associated to string or cycle hypergraphs. As an application, among these ideals we characterize all the Cohen-Macaulay ones.

Abstract. Let R be a polynomial ring over a field. We prove an upper bound for the multiplicity of R/I when I is a homogeneous ideal of the form I = J + (F ), where J is a Cohen-Macaulay ideal and F / ∈ J. The bound is given in terms of two invariants of R/J and the degree of F . We show that ideals achieving this upper bound have high depth, and provide a purely numerical criterion for the Cohen-Macaulay property. Applications to quasi-Gorenstein rings and almost complete intersections are given.

Motivated by a question of Stillman, we find a sharp upper bound for the projective dimension of ideals of height two generated by quadrics. In a polynomial ring with arbitrary large number of variables, we prove that ideals generated by n quadrics define cyclic modules with projective dimension at most 2n − 2. We refine this bound according to the multiplicity of the ideal. We ask whether tight upper bounds for the projective dimension of ideals generated by quadrics can be expressed only in terms of their height and number of minimal generators.

Star configurations of hypersurfaces are schemes in P n widely generalizing star configurations of points. Their rich structure allows them to be studied using tools from algebraic geometry, combinatorics, commutative algebra and representation theory. In particular, there has been much interest in understanding how "fattening" these schemes affects the algebraic properties of these configurations or, in other words, understanding the symbolic powers I (m) of their defining ideals I.In the present paper (1) we prove a structure theorem for I (m) , giving an explicit description of a minimal generating set of I (m) (overall, and in each degree) which also yields a minimal generating set of the module I (m) /I m -which measures how far is I (m) from I m . These results are new even for monomial star configurations or star configurations of points; (2) we introduce a notion of ideals with c.i. quotients, generalizing ideals with linear quotients, and show that I (m) have c.i. quotients. As a corollary we obtain that symbolic powers of ideals of star configurations of points have linear quotients; (3) we find a general formula for all graded Betti numbers of I (m) ; (4) we prove that a little bit more than the bottom half of the Betti table of I (m) has a regular, almost hypnotic, pattern, and provide a simple closed formula for all these graded Betti numbers and the last irregular strand in the Betti table.Other applications include improving and widely extending results by Galetto, Geramita, Shin and Van Tuyl, and providing explicit new general formulas for the minimal number of generators and the symbolic defects of star configurations.Inspired by Young tableaux, we introduce a "canonical" way of writing any monomial in any given set of polynomials, which may be of independent interest. We prove its existence and uniqueness under fairly general assumption. Along the way, we exploit a connection between the minimal generators G (m) of I (m) and positive solutions to Diophantine equations, and a connection between G (m) and partitions of m via the canonical form of monomials. Our methods are characteristic-free.