Sharp upper bounds of the Betti numbers for a given Hilbert polynomial

We show that there exists a saturated graded ideal in a standard graded polynomial ring which has the largest total Betti numbers among all saturated graded ideals for a fixed Hilbert polynomial.Comment: 43 page

Generic and Cogeneric Monomial Ideals

Monomial ideals which are generic with respect to either their generators or irreducible components have minimal free resolutions derived from simplicial complexes. For a generic monomial ideal, the associated primes satisfy a saturated chain condition, and the Cohen-Macaulay property implies shellability for both the Scarf complex and the Stanley-Reisner complex. Reverse lexicographic initial ideals of generic lattice ideals are generic. Cohen-Macaulayness for cogeneric ideals is characterized combinatorially; in the cogeneric case the Cohen-Macaulay type is greater than or equal to the number of irreducible components. Methods of proof include Alexander duality and Stanley's theory of local h-vectors.Comment: 15 pages, LaTe

HILBERT POLYNOMIALS AND STRONGLY STABLE IDEALS

Strongly stable ideals are important in algebraic geometry, commutative algebra, and combinatorics. Prompted, for example, by combinatorial approaches for studying Hilbert schemes and the existence of maximal total Betti numbers among saturated ideals with a given Hilbert polynomial, three algorithms are presented. Each of these algorithms produces all strongly stable ideals with some prescribed property: the saturated strongly stable ideals with a given Hilbert polynomial, the almost lexsegment ideals with a given Hilbert polynomial, and the saturated strongly stable ideals with a given Hilbert function. Bounds for the complexity of our algorithms are included. Also included are some applications for these algorithms and some estimates for counting strongly stable ideals with a fixed Hilbert polynomial

Segments and Hilbert schemes of points

Using results obtained from the study of homogeneous ideals sharing the same initial ideal with respect to some term order, we prove the singularity of the point corresponding to a segment ideal with respect to the revlex term order in the Hilbert scheme of points in $\mathbb{P}^n$. In this context, we look inside properties of several types of "segment" ideals that we define and compare. This study led us to focus our attention also to connections between the shape of generators of Borel ideals and the related Hilbert polynomial, providing an algorithm for computing all saturated Borel ideals with the given Hilbert polynomial.Comment: 19 pages, 2 figures. Comments and suggestions are welcome

Strongly stable ideals and Hilbert polynomials

The \texttt{StronglyStableIdeals} package for \textit{Macaulay2} provides a method to compute all saturated strongly stable ideals in a given polynomial ring with a fixed Hilbert polynomial. A description of the main method and auxiliary tools is given.Comment: Source code available as an ancillary file. Final versio

Multiplicity of the saturated special fiber ring of height two perfect ideals

Let $R$ be a polynomial ring and $I \subset R$ be a perfect ideal of height two minimally generated by forms of the same degree. We provide a formula for the multiplicity of the saturated special fiber ring of $I$. Interestingly, this formula is equal to an elementary symmetric polynomial in terms of the degrees of the syzygies of $I$. Applying ideas introduced in arXiv:1805.05180, we obtain the value of the j-multiplicity of $I$ and an effective method for determining the degree and birationality of rational maps defined by homogeneous generators of $I$.Comment: to appear in Proc. Amer. Math. So