4 citations · 12 across the 18 of their papers we have counts for
25 papers
A Linear Bound on the Projective Dimension of Height 3 Quadratic Ideals
Zachary Greif, Paolo Mantero, Jason McCullough
In 2016, Ananyan and Hochster gave the first proof of a positive answer to Stillman's Question, which asked for a bound on the projective dimension of a graded polynomial ideal pur…
Focal matroids of covers and homological properties of matroids
Paolo Mantero, Vinh Nguyen
In this paper we prove that the Stanley--Reisner ideal or cover ideal of a matroid is minimally resolvable by iterated mapping cones. As a technical tool for this purpose, we i…
Rational symbolic powers of ideals
Souvik Dey, Tai Huy Ha, Dipendranath Mahato +1
We introduce and study rational symbolic powers of ideals in Noetherian rings. We give membership criteria for rational symbolic powers and discuss settings where they agree with i…
Slightly mixed symbolic powers of matroids are locally glicci
Paolo Mantero, Vinh Nguyen
Let $\M$ be a matroid, and let $I_{\M}$ be either the Stanley--Reisner or the cover ideal of $\M$. In this paper we prove that for any matroid $\M$ on , any $\ell\in \ZZ_+$, a…
Ordinary and symbolic powers of matroids via polarization
Justin Lyle, Paolo Mantero
In this paper, we propose a uniform approach to tackle problems about squarefree monomial ideals whose powers have good properties. We employ this approach to achieve a twofold goa…
The Structure of Symbolic Powers of Matroids
Paolo Mantero, Vinh Nguyen
We describe the structure of the symbolic powers of the Stanley-Reisner ideals, and cover ideals, , of matroids. We (a) prove a structure theorem describing a minim…
A formula for symbolic powers
Paolo Mantero, Cleto B. Miranda-Neto, Uwe Nagel
Let be a Cohen-Macaulay ring which is local or standard graded over a field, and let be an unmixed ideal that is also generically a complete intersection. Our goal in this…
The structure of Koszul algebras defined by four quadrics
Paolo Mantero, Matthew Mastroeni
Avramov, Conca, and Iyengar ask whether for all when is a Koszul algebra minimally defined by quadrics. In recent work, we give an affi…
The Alexander-Hirschowitz theorem and related problems
Huy Tai Ha, Paolo Mantero
We present a proof of the celebrated result due to Alexander and Hirschowitz which determines when a general set of double points in has the expected Hilbert function…
Betti numbers of Koszul algebras defined by four quadrics
Paolo Mantero, Matthew Mastroeni
Let be an ideal generated by quadrics in a standard graded polynomial ring over a field. A question of Avramov, Conca, and Iyengar asks whether the Betti numbers of $R = S/…
The structure and free resolutions of the symbolic powers of star configurations of hypersurfaces
Paolo Mantero
Star configurations of points are configurations with known (and conjectured) extremal behaviors among all configurations of points in ; additional interest come fro…
Singularities of Rees-like Algebras
Paolo Mantero, Jason McCullough, Lance Edward Miller
Recently, Peeva and the second author constructed irreducible projective varieties with regularity much larger than their degree, yielding counterexamples to the Eisenbud-Goto Conj…
The projective dimension of three cubics is at most 5
Paolo Mantero, Jason McCullough
Let be a polynomial ring over a field and an ideal generated by three forms of degree three. Motivated by Stillman's question, Engheta proved that the projective dimension…
Chudnovsky's Conjecture for very general points in
Louiza Fouli, Paolo Mantero, Yu Xie
We prove a long-standing conjecture of Chudnovsky for very general and generic points in , where is an algebraically closed field of characteristic zero, and fo…
Hypergraphs with high projective dimension and 1-dimensional Hypergraphs
Kuei-Nuan Lin, Paolo Mantero
We prove a sufficient and a necessary condition for a square-free monomial ideal associated to a (dual) hypergraph to have projective dimension equal to the minimal number of g…
Ubiquity of complete intersection liaison classes
Mark Johnson, Paolo Mantero
In this paper, we provide constructions to enumerate large numbers of CI-liaison classes. To this end, we introduce a liaison invariant and prove several results concerning it, not…
Arithmetical rank of strings and cycles
Kyouko Kimura, Paolo Mantero
Let be a polynomial ring over a field . To a given squarefree monomial ideal , one can associate a hypergraph . In this article, we prove that the arithme…
A Tight Bound on the Projective Dimension of Four Quadrics
Craig Huneke, Paolo Mantero, Jason McCullough +1
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…
A multiplicity bound for graded rings and a criterion for the Cohen-Macaulay property
Craig Huneke, Paolo Mantero, Jason McCullough +1
Let be a polynomial ring over a field. We prove an upper bound for the multiplicity of when is a homogeneous ideal of the form , where is a Cohen-Macaula…
Generalized stretched ideals and Sally Conjecture
Paolo Mantero, Yu Xie
We introduce the concept of -stretched ideals in a Noetherian local ring. This notion generalizes to arbitrary ideals the classical notion of stretched -primary id…
Projective Dimension of String and Cycle Hypergraphs
Kuei-Nuan Lin, Paolo Mantero
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 a…
The projective dimension of codimension two algebra presented by quadrics
Craig Huneke, Paolo Mantero, Jason McCullough +1
We prove 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.
Multiple Structures with Arbitrarily Large Projective Dimension on Linear Subspaces
Craig Huneke, Paolo Mantero, Jason McCullough +1
Let be an algebraically closed field. There has been much interest in characterizing multiple structures in defined on a linear subspace of small codimension under addi…
Generalized stretched ideals and Sally's Conjecture
Paolo Mantero, Yu Xie
Given a finite module over a Noetherian local ring $(R, \m)$, we introduce the concept of -stretched ideals on . Thanks to a crucial specialization lemma, we show that th…