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Jorge Martín-Morales

Publications and source records attributed to Jorge Martín-Morales.

At least 19 recordsLinked to original sources
arXiv2026

On the poles of zeta functions for finite morphisms between normal surfaces

For a divisor representing a function and another divisor representing a differential form on a normal surface singularity, there is a notion of motivic and topological zeta function. In this paper, given a finite morphism between two normal surfaces, we prove that the set of poles of the motivic zeta function associated with the target is contained in the one associated with the source. We illustrate by examples that this inclusion is strict in general, and that on the topological level there are in general no inclusions between the sets of poles on source and target. On the other hand, when the morphism is the quotient map induced by an action of a finite abelian group on $\mathbb{C}^2$, and the divisor associated with the differential form on the source is trivial, we do show equality between the corresponding sets of poles, both on motivic and topological level. In addition, again for the quotient map induced by an action of a finite abelian group on $\mathbb{C}^2$, but now with a general divisor associated with a differential form, we provide a criterion when the topological zeta function on the target is just a multiple of the one on the source. Finally, we compare log canonical models on source and target with a view on zeta functions.

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arXiv2026

Modules of logarithmic derivations in weighted projective spaces and applications to free divisors

We introduce a weighted version of the module of logarithmic derivations of a divisor in weighted projective space, and provide a generalization of Saito's criterion for freeness in terms of weighted multiple eigenschemes (wME-schemes). Freeness of the nonstandard Z-graded module allows one to consider big families of free divisors in affine and standard projective space, i.e. when the module of logarithmic derivations of the divisor is free over the respective coordinate rings. We present a method to identify and construct these new families of free divisors in affine and projective space in any dimension, and give numerous explicit examples.

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arXiv2026

An algorithm for annihilator and Bernstein-Sato polynomial of a rational function

The singularity theory of rational functions, i.e., the quotient of two polynomials, has been investigated in the past two decades. The Bernstein-Sato polynomial of a rational function has recently been introduced by Takeuchi. However, only trivial examples are known. We provide an algorithm for computing the Bernstein-Sato polynomial in this context. The strategy is to compute the annihilator of the rational function by using the annihilator of the pair consisting of the numerator and denominator of the quotient. In a natural way a non-vanishing condition on the Bernstein-Sato ideal of the pair appears. This method has been implemented in freely available computer algebra system SINGULAR. It relies on Gröbner bases in noncommutative PBW algebras. The algorithm allows us to exhibit some explicit non-trivial examples and to support some existing conjectures.

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arXiv2024

Counting points with Riemann-Roch formulas

We provide an algorithm for computing the number of integral points lying in certain triangles that do not have integral vertices. We use techniques from Algebraic Geometry such as the Riemann-Roch formula for weighted projective planes and resolution of singularities. We analyze the complexity of the method and show that the worst case is given by the Fibonacci sequence. At the end of the manuscript a concrete example is developed in detail where the interplay with other invariants of singularity theory is also treated.

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arXiv2023

Cyclic coverings of rational normal surfaces which are quotients of a product of curves

This paper deals with cyclic covers of a large family of rational normal surfaces that can also be described as quotients of a product, where the factors are cyclic covers of algebraic curves. We use a generalization of Esnault-Viehweg method to show that the action of the monodromy on the first Betti group of the covering (and its Hodge structure) splits as a direct sum of the same data for some specific cyclic covers over $\mathbb{P}^1$. This has applications to the study of Lê-Yomdin surface singularities, in particular to the action of the monodromy on the Mixed Hodge Structure, as well as to isotrivial fibered surfaces.

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arXiv2020

The monodromy conjecture for a space monomial curve with a plane semigroup

This article investigates the monodromy conjecture for a space monomial curve that appears as the special fiber of an equisingular family of curves with a plane branch as generic fiber. Roughly speaking, the monodromy conjecture states that every pole of the motivic, or related, Igusa zeta function induces an eigenvalue of monodromy. As the poles of the motivic zeta function associated with such a space monomial curve have been determined in earlier work, it remains to study the eigenvalues of monodromy. After reducing the problem to the curve seen as a Cartier divisor on a generic embedding surface, we construct an embedded $\mathbb Q$-resolution of this pair and use an A'Campo formula in terms of this resolution to compute the zeta function of monodromy. Combining all results, we prove the monodromy conjecture for this class of monomial curves.

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arXiv2020

Normal surface singularities with an integral homology sphere link related to space monomial curves with a plane semigroup

In this article, we consider an infinite family of normal surface singularities with an integral homology sphere link which is related to the family of space monomial curves with a plane semigroup. These monomial curves appear as the special fibers of equisingular families of curves whose generic fibers are a complex plane branch, and the related surface singularities appear in a proof of the monodromy conjecture for these curves. To investigate whether the link of a normal surface singularity is an integral homology sphere, one can use a characterization that depends on the determinant of the intersection matrix of a (partial) resolution. To study our family, we apply this characterization with a partial toric resolution of our singularities constructed as a sequence of weighted blow-ups.

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arXiv2020

Local invariants of minimal generic curves on rational surfaces

Let (C,0) be a reduced curve germ in a normal surface singularity (X,0). The main goal is to recover the delta invariant of the abstract curve (C,0) from the topology of the embedding. We give explicit formulae whenever (C,0) is minimal generic and (X,0) is rational (as a continuation of previous works of the authors). Additionally we prove that if (X,0) is a quotient singularity, then the delta invariant of (C,0) only admits the values r-1 or r, where r is the number or irreducible components of (C,0). (r-1 realizes the extremal lower bound, valid only for `ordinary r-tuples'.)

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arXiv2020

Motivic Zeta Functions on $\mathds{Q}$-Gorenstein Varieties

We study motivic zeta functions for $\mathds{Q}$-divisors in a $\mathds{Q}$-Gorenstein variety. By using a toric partial resolution of singularities we reduce this study to the local case of two normal crossing divisors where the ambient space is an abelian quotient singularity. For the latter we provide a closed formula which is worked out directly on the quotient singular variety. As a first application we provide a family of surface singularities where the use of weighted blow-ups reduces the set of candidate poles drastically. We also present an example of a quotient singularity under the action of a nonabelian group, from which we compute some invariants of motivic nature after constructing a $\mathds{Q}$-resolution.

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arXiv2020

The delta invariant of curves on rational surfaces II: Poincaré series and topological aspects

In this article we study abstract and embedded invariants of reduced curve germs via topological techniques. One of the most important numerical analytic invariants of an abstract curve is its delta invariant. Our primary goal is to develop delta invariant formulae for curves embedded in rational singularities in terms of embedded data. The topological machinery not only produces formulae, but it also creates deep connections with the theory of (analytical and topological) multivariable Poincaré series.

math.GT↗
arXiv2020

Cyclic branched coverings of surfaces with abelian quotient singularities

Esnault-Viehweg developed the theory of cyclic branched coverings $\tilde X\to X$ of smooth surfaces providing a very explicit formula for the decomposition of $H^1(\tilde X,\mathbb{C})$ in terms of a resolution of the ramification locus. Later, the first author applies this to the particular case of coverings of $\mathbb{P}^2$ reducing the problem to a combination of global and local conditions on projective curves. In this paper we extend the above results in three directions: first, the theory is extended to surfaces with quotient singularities, second the ramification locus can be partially resolved and need not be reduced, and finally global and local conditions are given to describe the irregularity of cyclic branched coverings of the weighted projective plane. The techniques required for these results are conceptually different and provide simpler proofs for the classical results. For instance, the local contribution comes from certain modules that have the flavor of quasi-adjunction and multiplier ideals on singular surfaces. As an application, a Zariski pair of curves on a singular surface is described. In particular, we prove the existence of two cuspidal curves of degree 12 in the weighted projective plane $\mathbb{P}^2_{(1,1,3)}$ with the same singularities but non-homeomorphic embeddings. This is shown by proving that the cyclic covers of $\mathbb{P}^2_{(1,1,3)}$ of order 12 ramified along the curves have different irregularity. In the process, only a partial resolution of singularities is required.

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arXiv2019

Triangular curves and cyclotomic Zariski tuples

The purpose of this paper is to exhibit infinite families of conjugate projective curves in a number field whose complement have the same abelian fundamental group, but are non-homeomorphic. In particular, for any $d>3$ we find Zariski tuples parametrized by the $d$-roots of unity up to complex conjugation. As a consequence, for any divisor $m$ of $d$, $m\neq 1,2,3,4,6$, we find arithmetic Zariski $\frac{ϕ(m)}{2}$-tuples with coefficients in the corresponding cyclotomic field. These curves have abelian fundamental group and they are distinguished using a linking invariant.

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arXiv2019

Delta invariant of curves on rational surfaces I. The analytic approach

We prove that if (C,0) is a reduced curve germ on a rational surface singularity (X,0) then its delta invariant can be recovered by a concrete expression associated with the embedded topological type of the pair (X,C). Furthermore, we also identify it with another (a priori) embedded analytic invariant, which is motivated by the theory of adjoint ideals. Finally, we connect our formulae with the local correction term at singular points of the global Riemann--Roch formula, valid for projective normal surfaces, introduced by Blache.

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arXiv2018

Coverings of rational ruled normal surfaces

In this work we use arithmetic, geometric, and combinatorial techniques to compute the cohomology of Weil divisors of a special class of normal surfaces, the so-called rational ruled toric surfaces. These computations are used to study the topology of cyclic coverings of such surfaces ramified along Q-normal crossing divisors.

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arXiv2017

Wirtinger curves, Artin groups, and hypocycloids

The computation of the fundamental group of the complement of an algebraic plane curve has been theoretically solved since Zariski-van Kampen, but actual computations are usually cumbersome. In this work, we describe the notion of Wirtinger presentation of such a group relying on the real picture of the curve and with the same combinatorial flavor as the classical Wirtinger presentation; we determine a significant family of curves for which Wirtinger presentation provides the required fundamental group. The above methods allow us to compute that fundamental group for an infinite subfamily of hypocycloids, relating them with Artin groups.

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arXiv2015

An algebraic approach to the minimum-cost multi-impulse orbit transfer problem

We present a purely algebraic formulation (i.e. polynomial equations only) of the minimum-cost multi-impulse orbit transfer problem without time constraints, while keeping all the variables with a precise physical meaning. We apply general algebraic techniques to solve these equations (resultants, Gröbner bases, etc.) in several situations of practical interest of different degrees of generality. For instance, we provide a proof of the optimality of the Hohmann transfer for the minimum fuel 2-impulse circular to circular orbit transfer problem, and we provide a general formula for the optimal 2-impulse in-plane transfer between two rotated elliptical orbits under a mild symmetry assumption on the two points where the impulses are applied (which we conjecture that can be removed).

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arXiv2015

Cartier and Weil Divisors on Varieties with Quotient Singularities

The main goal of this paper is to show that the notions of Weil and Cartier $\mathbb{Q}$-divisors coincide for $V$-manifolds and give a procedure to express a rational Weil divisor as a rational Cartier divisor. The theory is illustrated on weighted projective spaces and weighted blow-ups.

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