On the multiplicity of tangent cones of monomial curves

Let $\Lambda$ be a numerical semigroup, $\mathcal{C}\subseteq \mathbb{A}^n$ the monomial curve singularity associated to $\Lambda$, and $\mathcal{T}$ its tangent cone. In this paper we provide a sharp upper bound for the least positive integer in $\Lambda$ in terms of the codimension and the maximum degree of the equations of $\mathcal{T}$, when $\mathcal{T}$ is not a complete intersection. A special case of this result settles a question of J. Herzog and D. Stamate.Comment: To appear on Arkiv f\"or Matemati

s-Hankel hypermatrices and 2 x 2 determinantal ideals

We introduce the concept of s-Hankel hypermatrix, which generalizes both Hankel matrices and generic hypermatrices. We study two determinantal ideals associated to an s-Hankel hypermatrix: the ideal I generated by certain 2 x 2 slice minors, and the ideal \tilde{I} generated by certain 2 x 2 generalized minors. We describe the structure of these two ideals, with particular attention to the primary decomposition of I, and provide the explicit list of minimal primes for large values of s. Finally we give some geometrical interpretations and generalise a theorem of Watanabe

On a conjecture of Wilf about the Frobenius number

Given coprime positive integers $a_1 < ...< a_d$, the Frobenius number $F$ is the largest integer which is not representable as a non-negative integer combination of the $a_i$. Let $g$ denote the number of all non-representable positive integers: Wilf conjectured that $d \geq \frac{F+1}{F+1-g}$. We prove that for every fixed value of $\lceil \frac{a_1}{d} \rceil$ the conjecture holds for all values of $a_1$ which are sufficiently large and are not divisible by a finite set of primes. We also propose a generalization in the context of one-dimensional local rings and a question on the equality $d = \frac{F+1}{F+1-g}$

Syzygies in Hilbert schemes of complete intersections

Let $d_1, \ldots, d_{c}$ be positive integers and let $Y \subseteq \mathbb{P}^n$ be the monomial complete intersection defined by the vanishing of $x_1^{d_1}, \ldots, x_{c}^{d_{c}}$. For each Hilbert polynomial $p(\zeta)$ we construct a distinguished point in the Hilbert scheme $\mathrm{Hilb}^{p(\zeta)}(Y)$, called the expansive point. We develop a theory of expansive ideals, and show that they play for Hilbert polynomials the same role lexicographic ideals play for Hilbert functions. For instance, expansive ideals maximize number of generators and syzygies, they form descending chains of inclusions, and exhibit an extremal behavior with respect to hyperplane sections. Conjecturally, expansive subschemes provide uniform sharp upper bounds for the syzygies of subschemes $Z \in \mathrm{Hilb}^{p(\zeta)}(X)$ for all complete intersections $X = X(d_1, \ldots, d_{c}) \subseteq \mathrm{P}^{n}$. In some cases, the expansive point achieves extremal Betti numbers for the infinite free resolutions associated to subschemes in $\mathrm{Hilb}^{p(\zeta)}(Y)$. Our approach is new even in the special case $Y = \mathbb{P}^{n}$, where it provides several novel results and a simpler proof of a theorem of Murai and the first author.Comment: 26 page

On the Lex-plus-powers Conjecture

Let $S$ be a polynomial ring over a field and $I\subseteq S$ a homogeneous ideal containing a regular sequence of forms of degrees $d_1, \ldots, d_c$. In this paper we prove the Lex-plus-powers Conjecture when the field has characteristic 0 for all regular sequences such that $d_i \geq \sum_{j=1}^{i-1} (d_j-1)+1$ for each $i$; that is, we show that the Betti table of $I$ is bounded above by the Betti table of the lex-plus-powers ideal of $I$. As an application, when the characteristic is 0, we obtain bounds for the Betti numbers of any homogeneous ideal containing a regular sequence of known degrees, which are sharper than the previously known ones from the Bigatti-Hulett-Pardue Theorem.Comment: To appear in Advances in Mathematic

Correspondence scrolls

This paper initiates the study of a class of schemes that we call correspondence scrolls, which includes the rational normal scrolls and linearly embedded projective bundle of decomposable bundles, as well as degenerate K3 surfaces, Calabi-Yau 3-folds, and many other examples

On the growth of deviations

The deviations of a graded algebra are a sequence of integers that determine the Poincare series of its residue field and arise as the number of generators of certain DG algebras. In a sense, deviations measure how far a ring is from being a complete intersection. In this paper we study extremal deviations among those of algebras with a fixed Hilbert series. In this setting, we prove that, like the Betti numbers, deviations do not decrease when passing to an initial ideal and are maximized by the Lex-segment ideal. We also prove that deviations grow exponentially for Golod rings and for certain quadratic monomial algebras.Comment: Corrected some minor typos in the version published in PAM

Edge ideals and DG algebra resolutions

Let $R= S/I$ where $S=k[T_1, \ldots, T_n]$ and $I$ is a homogeneous ideal in $S$. The acyclic closure $R \langle Y \rangle$ of $k$ over $R$ is a DG algebra resolution obtained by means of Tate's process of adjoining variables to kill cycles. In a similar way one can obtain the minimal model $S[X]$, a DG algebra resolution of $R$ over $S$. By a theorem of Avramov there is a tight connection between these two resolutions. In this paper we study these two resolutions when $I$ is the edge ideal of a path or a cycle. We determine the behavior of the deviations $\varepsilon_i(R)$, which are the number of variables in $R\langle Y \rangle$ in homological degree $i$. We apply our results to the study of the $k$-algebra structure of the Koszul homology of $R$