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Julio José Moyano-Fernández

Publications and source records attributed to Julio José Moyano-Fernández.

At least 37 records · Page 2Linked to original sources
arXiv2017

Newton-Okounkov bodies of exceptional curve valuations

We prove that the Newton-Okounkov body of the flag $E_{\bullet}:= \left\{ X=X_r \supset E_r \supset \{q\} \right\}$, defined by the surface $X$ and the exceptional divisor $E_r$ given by any divisorial valuation of the complex projective plane $\mathbb{P}^2$, with respect to the pull-back of the line-bundle $\mathcal{O}_{\mathbb{P}^2} (1)$ is either a triangle or a quadrilateral, characterizing when it is a triangle or a quadrilateral. We also describe the vertices of that figure. Finally, we introduce a large family of flags for which we determine explicitly their Newton-Okounkov bodies which turn out to be triangular.

math.AG↗
arXiv2017

Minimal plane valuations

We consider the last value $\hatμ (ν)$ of the vanishing sequence of $H^0(L)$ along a divisorial or irrational valuation $ν$ centered at $\mathcal{O}_{\mathbb{P}^2,p}$, where $L$ resp. $p$ is a line resp. a point of the projective plane $\mathbb{P}^2$ over an algebraically closed field. This value contains, for valuations, similar information as that given by Seshadri constants for points. It is always true that $\hatμ (ν) \geq \sqrt{1 / \mathrm{vol}(ν)}$ and minimal valuations are those satisfying the equality. In this paper, we prove that the Greuel-Lossen-Shustin Conjecture implies a variation of the Nagata Conjecture involving minimal valuations (that extends the one stated in the paper "Very general monomial valuations of $\mathbb{P}^2$ and a Nagata type conjecture" by Dumnicki et al. to the whole set of divisorial and irrational valuations of the projective plane) which also implies the original one. We also provide infinitely many families of very general minimal valuations with an arbitrary number of Puiseux exponents, and an asymptotic result that can be considered as an evidence in the direction of the mentioned conjecture by Dumnicki et al.

math.AG↗
arXiv2017

Stanley depth and the lcm-lattice

In this paper we show that the Stanley depth, as well as the usual depth, are essentially determined by the lcm-lattice. More precisely, we show that for quotients $I/J$ of monomial ideals $J\subset I$, both invariants behave monotonic with respect to certain maps defined on their lcm-lattice. This allows simple and uniform proofs of many new and known results on the Stanley depth. In particular, we obtain a generalization of our result on polarization presented in the reference [IKMF14]. We also obtain a useful description of the class of all monomial ideals with a given lcm-lattice, which is independent from our applications to the Stanley depth.

math.AC↗
arXiv2016

Hilbert series of modules over positively graded polynomials rings

In this note, we give examples of formal power series satisfying certain conditions that cannot be realized as Hilbert series of finitely generated modules. This answers to the negative a question raised in a recent article by the second and the third author. On the other hand, we show that the answer is positive after multiplication with a scalar.

math.AC↗
arXiv2016

Which series are Hilbert series of graded modules over polynomial rings?

Let $S$ be a multigraded polynomial ring such that the degree of each variable is a unit vector; so $S$ is the homogeneous coordinate ring of a product of projective spaces. In this setting, we characterize the formal Laurent series which arise as Hilbert series of finitely generated $S$-modules. Also we provide necessary conditions for a formal Laurent series to be the Hilbert series of a finitely generated module with a given depth. In the bigraded case (corresponding to the product of two projective spaces), we completely classify the Hilbert series of finitely generated modules of positive depth.

math.AC↗
arXiv2016

Linear optics only allows every possible quantum operation for one photon or one port

We study the evolution of the quantum state of $n$ photons in $m$ different modes when they go through a lossless linear optical system. We show that there are quantum evolution operators $U$ that cannot be built with linear optics alone unless the number of photons or the number of modes is equal to one. The evolution for single photons can be controlled with the known realization of any unitary proved by Reck, Zeilinger, Bernstein and Bertani. The evolution for a single mode corresponds to the trivial evolution in a phase shifter. We analyze these two cases and prove that any other combination of the number of photons and modes produces a Hilbert state too big for the linear optics system to give any desired evolution.

quant-ph↗
arXiv2015

How to compute the Stanley depth of a module

In this paper we introduce an algorithm for computing the Stanley depth of a finitely generated multigraded module $M$ over the polynomial ring $\mathbb{K}[X_1, \ldots, X_n]$. As an application, we give an example of a module whose Stanley depth is strictly greater than the depth of its syzygy module. In particular, we obtain complete answers for two open questions raised by Herzog. Moreover, we show that the question whether $M$ has Stanley depth at least $r$ can be reduced to the question whether a certain combinatorially defined polytope $\mathscr{P}$ contains a $\mathbb{Z}^n$-lattice point.

math.AC↗
arXiv2015

Lcm-lattices and Stanley depth: a first computational approach

Let $\mathbb{K}$ be a field, and let $S=\mathbb{K}[X_1, ..., X_n]$ be the polynomial ring. Let $I$ be a monomial ideal of $S$ with up to 5 generators. In this paper, we present a computational experiment which allows us to prove that $\mathrm{depth}_S S/I = \mathrm{sdepth}_S S/I < \mathrm{sdepth}_S I$. This shows that the Stanley conjecture is true for $S/I$ and $I$, if $I$ can be generated by at most 5 monomials. The result also brings additional computational evidence for a conjecture made by Herzog.

math.AC↗
arXiv2014

The behavior of Stanley depth under polarization

Let $K$ be a field, $R=K[X_1, ..., X_n]$ be the polynomial ring and $J \subsetneq I$ two monomial ideals in $R$. In this paper we show that $\mathrm{sdepth}\ {I/J} - \mathrm{depth}\ {I/J} = \mathrm{sdepth}\ {I^p/J^p}-\mathrm{depth}\ {I^p/J^p}$, where $\mathrm{sdepth}\ I/J$ denotes the Stanley depth and $I^p$ denotes the polarization. This solves a conjecture by Herzog and reduces the famous Stanley conjecture (for modules of the form $I/J$) to the squarefree case. As a consequence, the Stanley conjecture for algebras of the form $R/I$ and the well-known combinatorial conjecture that every Cohen-Macaulay simplicial complex is partitionable are equivalent.

math.AC↗
arXiv2013

On multi-index filtrations associated to Weierstrass semigroups

The aim of this paper is to review the main techniques in the computation of Weierstraßsemigroup at several points of curves defined over perfect fields, with special emphasis on the case of two points. Some hints about the usage of some packages of the computer algebra software \textsc{Singular} are also given.

math.AG↗
arXiv2013

Duality and syzygies for semimodules over numerical semigroups

Let $Γ=\langle α, β\rangle$ be a numerical semigroup. In this article we consider the dual $Δ^*$ of a $Γ$-semimodule $Δ$; in particular we deduce a formula that expresses the minimal set of generators of $Δ^*$ in terms of the generators of $Δ$. As applications we compute the minimal graded free resolution of a graded $\mathbb{F}[t^α,t^β]$-submodule of $\mathbb{F}[t]$, and we investigate the structure of the selfdual $Γ$-semimodules, leading to a new way of counting them.

math.CO↗
arXiv2013

How to compute the multigraded Hilbert depth of a module

The aim of this paper is to introduce a method for computing Hilbert decompositions (and consequently the Hilbert depth) of a finitely generated multigraded module $M$ over the polynomial ring $K[X_1,..., X_n]$ by reducing the problem to the computation of the finite set of the new defined Hilbert partitions. Moreover, in the last section, we show that Hilbert partitions may also be used for computing the Stanley depth of the module $M$.

math.AC↗
arXiv2013

Lattice paths with given number of turns and semimodules over numerical semigroups

Let Γ=<α, β> be a numerical semigroup. In this article we consider several relations between the so-called Γ-semimodules and lattice paths from (0,α) to (β,0): we investigate isomorphism classes of Γ-semimodules as well as certain subsets of the set of gaps of Γ, and finally syzygies of Γ-semimodules. In particular we compute the number of Γ-semimodules which are isomorphic with their k-th syzygy for some k.

math.CO↗
arXiv2013

Hilbert regularity of ZZ-graded modules over polynomial rings

Let M be a finitely generated ZZ-graded module over the standard graded polynomial ring R=K[X_1, ..., X_n] with K a field, and let H_M(t)=Q_M(t)/(1-t)^d be the Hilbert series of M. We introduce the Hilbert regularity of M as the lowest possible value of the Castelnuovo-Mumford regularity for an R-module with Hilbert series H_M. Our main result is an arithmetical description of this invariant which connects the Hilbert regularity of M to the smallest k such that the power series Q_M(1-t)/(1-t)^k has no negative coefficients. Finally we give an algorithm for the computation of the Hilbert regularity and the Hilbert depth of an R-module.

math.AC↗
arXiv2012

Hilbert depth of graded modules over polynomial rings in two variables

In this article we mainly consider the positively Z-graded polynomial ring R=F[X,Y] over an arbitrary field F and Hilbert series of finitely generated graded R-modules. The central result is an arithmetic criterion for such a series to be the Hilbert series of some R-module of positive depth. In the generic case, that is, the degrees of X and Y being coprime, this criterion can be formulated in terms of the numerical semigroup generated by those degrees.

math.AC↗
arXiv2011

Poincaré series for plane curve singularities and their behaviour under projections

Our purpose is to investigate all defined Poincaré series associated with multi-index filtrations and value semigroups of curve singularities---not necessarily complex---with regard to the property of forgetting variables, i.e., by making variables of the series to be 1. Generalised Poincaré series of motivic nature will be also considered.

math.AG↗