arXiv subjects

Aron Simis

Publications and source records attributed to Aron Simis.

49 records · Page 3Linked to original sources

arXiv2013

Symbolic powers of perfect ideals of codimension 2 and birational maps

This work is about symbolic powers of codimension two perfect ideals in a standard polynomial ring over a field, where the entries of the corresponding presentation matrix are general linear forms. The main contribution of the present approach is the use of the birational theory underlying the nature of the ideal and the details of a deep interlacing between generators of its symbolic powers and the inversion factors stemming from the inverse map to the birational map defined by the linear system spanned by the generators of this ideal. A full description of the corresponding symbolic Rees algebra is given in some cases.

math.AC↗

arXiv2012

Syzygies of differentials of forms

Given a standard graded polynomial ring $R=k[x_1,...,x_n]$ over a field $k$ of characteristic zero and a graded $k$-subalgebra $A=k[f_1,...,f_m]\subset R$, one relates the module $Ω_{A/k}$ of Kähler $k$-differentials of $A$ to the transposed Jacobian module $\mathcal{D}\subset \sum_{i=1}^n R dx_i$ of the forms $f_1,...,f_m$ by means of a {\em Leibniz map} $Ω_{A/k}\rar \mathcal{D}$ whose kernel is the torsion of $Ω_{A/k}$. Letting $\fp$ denote the $R$-submodule generated by the (image of the) syzygy module of $Ω_{A/k}$ and $\fz$ the syzygy module of $\mathcal{D}$, there is a natural inclusion $\fp\subset \fz$ coming from the chain rule for composite derivatives. The main goal is to give means to test when this inclusion is an equality -- in which case one says that the forms $f_1,...,f_m$ are {\em polarizable}. One surveys some classes of subalgebras that are generated by polarizable forms. The problem has some curious connections with constructs of commutative algebra, such as the Jacobian ideal, the conormal module and its torsion, homological dimension in $R$ and syzygies, complete intersections and Koszul algebras. Some of these connections trigger questions which have interest in their own.

math.AC↗

arXiv2012

New Constructions of Cremona Maps

One defines two ways of constructing rational maps derived from other rational maps, in a characteristic-free context. The first introduces the Newton complementary dual of a rational map. One main result is that this dual preserves birationality and gives an involutional map of the Cremona group to itself that restricts to the monomial Cremona subgroup and preserves de Jonquières maps. In the monomial restriction this duality commutes with taking inverse in the group, but is a not a group homomorphism. The second construction is an iterative process to obtain rational maps in increasing dimension. Starting with birational maps, it leads to rational maps whose topological degree is under control. Making use of monoids, the resulting construct is in fact birational if the original map is so. A variation of this idea is considered in order to preserve properties of the base ideal, such as Cohen--Macaulayness. Combining the two methods, one is able to produce explicit infinite families of Cohen--Macaulay Cremona maps with prescribed dimension, codimension and degree.

math.AC↗

arXiv2012

Extremal Rees Algebras

We study almost complete intersections ideals whose Rees algebras are extremal in the sense that some of their fundamental metrics---depth or relation type---have maximal or minimal values in the class. The focus is on those ideals that lead to almost Cohen--Macaulay algebras and our treatment is wholly concentrated on the nonlinear relations of the algebras. Several classes of such algebras are presented, some of a combinatorial origin. We offer a different prism to look at these questions with accompanying techniques. The main results are effective methods to calculate the invariants of these algebras.

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arXiv2012

Homology of Homogeneous Divisors

One deals with arbitrary reduced free divisors in a polynomial ring over a field of characteristic zero, by stressing the ideal theoretic and homological behavior of the corresponding singular locus. A particular emphasis is given to both weighted homogeneous and homogeneous polynomials, allowing to introduce new families of free divisors which do not come from hyperplane arrangements nor as explicit discriminants from singularity theory.

math.AC↗

arXiv2012

Implicitization of de Jonquières parametrizations

One introduces a class of projective parameterizations that resemble generalized de Jonquières maps. Any such parametrization defines a birational map $\mathfrak{F}$ of $\pp^n$ onto a hypersurface $V(F)\subset \pp^{n+1}$ with a strong handle to implicitization. From this side, the theory here developed extends recent work of Ben\ii tez--D'Andrea on monoid parameterizations. The paper deals with both ideal theoretic and effective aspects of the problem. The ring theoretic development gives information on the Castelnuovo--Mumford regularity of the base ideal of $\mathfrak{F}$. From the effective side, one gives an explicit formula of $°(F)$ involving data from the inverse map of $\mathfrak{F}$ and show how the present parametrization relates to monoid parameterizations.

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arXiv2011

Combinatorics of Cremona monomial maps

One studies Cremona monomial maps by combinatorial means. Among the results is a simple integer matrix theoretic proof that the inverse of a Cremona monomial map is also defined by monomials of fixed degree, and moreover, the set of monomials defining the inverse can be obtained explicitly in terms of the initial data. A neat consequence is drawn for the plane Cremona monomial group, in particular the known result saying that a plane Cremona (monomial) map and its inverse have the same degree. Included is a discussion about the computational side and/or implementation of the combinatorial invariants stemming from these questions.

math.AG↗

arXiv2011

The Aluffi Algebra

We deal with the quasi-symmetric algebra introduced by Paolo Aluffi, here named (embedded) Aluffi algebra. The algebra is a sort of "intermediate" algebra between the symmetric algebra and the Rees algebra of an ideal, which serves the purpose of introducing the characteristic cycle of a hypersurface in intersection theory. The results described in the present paper have an algebraic flavor and naturally connect with various themes of commutative algebra, such as standard bases á la Hironaka, Artin--Rees like questions, Valabrega--Valla ideals, ideals of linear type, relation type and analytic spread. We give estimates for the dimension of the Aluffi algebra and show that, pretty generally, the latter is equidimensional whenever the base ring is a hypersurface ring. There is a converse to this under certain conditions that essentially subsume the setup in Aluffi's theory, thus suggesting that this algebra will not handle cases other than the singular locus of a hypersurface. The torsion and the structure of the minimal primes of the algebra are clarified. In the case of a projective hypersurface the results are more precise and one is naturally led to look at families of projective plane singular curves to understand how the property of being of linear type deforms/specializes for the singular locus of a member. It is fairly elementary to show that the singular locus of an irreducible curve of degree at most 3 is of linear type. This is roundly false in degree larger than 4 and the picture looks pretty wild as we point out by means of some families. Degree 4 is the intriguing case. Here we are able to show that the singular locus of the generic member of a family of rational quartics, fixing the singularity type, is of linear type. We conjecture that every irreducible quartic has singular locus of linear type.

math.AC↗

arXiv2011

Cremona maps defined by monomials

Cremona maps defined by monomials of degree 2 are thoroughly analyzed and classified via integer arithmetic and graph combinatorics. In particular, the structure of the inverse map to such a monomial Cremona map is made very explicit as is the degree of its monomial defining coordinates. As a special case, one proves that any monomial Cremona map of degree 2 has inverse of degree 2 if and only if it is an involution up to permutation in the source and in the target. This statement is subsumed in a recent result of L. Pirio and F. Russo, but the proof is entirely different and holds in all characteristics. One unveils a close relationship binding together the normality of a monomial ideal, monomial Cremona maps and Hilbert bases of polyhedral cones. The latter suggests that facets of monomial Cremona theory may be NP-hard.

math.AC↗

arXiv2009

Elimination and nonlinear equations of Rees algebra

A new approach is established to computing the image of a rational map, whereby the use of approximation complexes is complemented with a detailed analysis of the torsion of the symmetric algebra in certain degrees. In the case the map is everywhere defined this analysis provides free resolutions of graded parts of the Rees algebra of the base ideal in degrees where it does not coincide with the corresponding symmetric algebra. A surprising fact is that the torsion in those degrees only contributes to the first free module in the resolution of the symmetric algebra modulo torsion. An additional point is that this contribution -- which of course corresponds to non linear equations of the Rees algebra -- can be described in these degrees in terms of non Koszul syzygies via certain upgrading maps in the vein of the ones introduced earlier by J. Herzog, the third named author and W. Vasconcelos. As a measure of the reach of this torsion analysis we could say that, in the case of a general everywhere defined map, half of the degrees where the torsion does not vanish are understood.

math.AC↗

arXiv2009

Equations of Almost Complete Intersections

In this paper we examine the role of four Hilbert functions in the determination of the defining relations of the Rees algebra of almost complete intersections of finite colength. Because three of the corresponding modules are Artinian, some of these relationships are very effective, opening up tracks to the determination of the equations and also to processes of going from homologically defined sets of equations to higher degrees ones assembled by resultants.

math.AC↗

arXiv2008

Polar syzygies in characteristic zero: the monomial case

Given a set of forms f={f_1,...,f_m} in R=k[x_1,...,x_n], where k is a field of characteristic zero, we focus on the first syzygy module Z of the transposed Jacobian module D(f), whose elements are called differential syzygies of f. There is a distinct submodule P of Z coming from the polynomial relations of f through its transposed Jacobian matrix, the elements of which are called polar syzygies of f. We say that f is polarizable if equality P=Z holds. This paper is concerned with the situation where f are monomials of degree 2, in which case one can naturally associate to them a graph G(f) with loops and translate the problem into a combinatorial one. A main result is a complete combinatorial characterization of polarizability in terms of special configurations in this graph. As a consequence, we show that polarizability implies normality of the subalgebra k[f] of R and that the converse holds provided the graph G(f) is free of certain degenerate configurations. One main combinatorial class of polarizability is the class of polymatroidal sets. We also prove that if the edge graph of G(f) has diameter at most 2 then f is polarizable. We establish a curious connection with birationality of rational maps defined by monomial quadrics.

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arXiv2006

Linear syzygies and birational combinatorics

Let $F$ be a finite set of monomials of the same degree $d\geq 2$ in a polynomial ring $R=k[x_1,...,x_n]$ over an arbitrary field $k$. We give some necessary and/or sufficient conditions for the birationality of the ring extension $k[F]\subset R^{(d)}$, where $R^{(d)}$ is the $d${\it th} Veronese subring of $R$. One of our results extends to arbitrary characteristic, in the case of rational monomial maps, a previous syzygy-theoretic birationality criterion in characteristic zero

math.AC↗