Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics

Frontmatter

Antonio Campillo

A Portrayal of His Life and Work

Abstract

The main purpose is to present a biographical portrayal of Antonio Campillo López. In addition to the most relevant aspects of his life and academic career, we offer a panoramic view of his scientific work in fields such as singularity theory, commutative algebra, algebraic geometry, or coding theory. We also stress that his endeavors have been closely linked to an important work of research training, from which a thriving school has emerged. Finally, his vision and influence will be considered through his numerous institutional and scientific policy responsibilities at all levels.

Francisco Monserrat, Sebastià Xambó-Descamps

Singularities in Positive Characteristic: Equisingularity, Classification, Determinacy

Abstract

In this survey paper we give an overview on some aspects of singularities of algebraic varieties over an algebraically closed field of arbitrary characteristic. We review in particular results on equisingularity of plane curve singularities, classification of hypersurface singularities and determinacy of arbitrary singularities. The section on equisingularity has its roots in two important early papers by Antonio Campillo. One emphasis is on the differences between positive and zero characteristic and on open problems.

Gert-Martin Greuel

Ultrametric Spaces of Branches on Arborescent Singularities

Abstract

Let S be a normal complex analytic surface singularity. We say that S is arborescent if the dual graph of any good resolution of it is a tree. Whenever A, B are distinct branches on S, we denote by A ⋅ B their intersection number in the sense of Mumford. If L is a fixed branch, we define U _L(A, B) = (L ⋅ A)(L ⋅ B)(A ⋅ B)⁻¹ when A ≠ B and U _L(A, A) = 0 otherwise. We generalize a theorem of Płoski concerning smooth germs of surfaces, by proving that whenever S is arborescent, then U _L is an ultrametric on the set of branches of S different from L. We compute the maximum of U _L, which gives an analog of a theorem of Teissier. We show that U _L encodes topological information about the structure of the embedded resolutions of any finite set of branches. This generalizes a theorem of Favre and Jonsson concerning the case when both S and L are smooth. We generalize also from smooth germs to arbitrary arborescent ones their valuative interpretation of the dual trees of the resolutions of S. Our proofs are based in an essential way on a determinantal identity of Eisenbud and Neumann.

Evelia R. García Barroso, Pedro D. González Pérez, Patrick Popescu-Pampu

Two Points of the Boundary of Toric Geometry

Abstract

This note presents two observations which have in common that they lie at the boundary of toric geometry. The first one because it concerns the deformation of affine toric varieties into non toric germs in order to understand how to avoid some ramification problems arising in the study of local uniformization in positive characteristic, and the second one because it uses limits of projective systems of equivariant birational maps of toric varieties to study the space of additive preorders on Z ^r for r ≥ 2.

Bernard Teissier

On the Milnor Formula in Arbitrary Characteristic

Abstract

The Milnor formula μ = 2δ − r + 1 relates the Milnor number μ, the double point number δ and the number r of branches of a plane curve singularity. It holds over the fields of characteristic zero. Melle and Wall based on a result by Deligne proved the inequality μ ≥ 2δ − r + 1 in arbitrary characteristic and showed that the equality μ = 2δ − r + 1 characterizes the singularities with no wild vanishing cycles. In this note we give an account of results on the Milnor formula in characteristic p. It holds if the plane singularity is Newton non-degenerate (Boubakri et al. Rev. Mat. Complut. 25:61–85, 2010) or if p is greater than the intersection number of the singularity with its generic polar (Nguyen Annales de l’Institut Fourier, Tome 66(5):2047–2066, 2016). Then we improve our result on the Milnor number of irreducible singularities (Bull. Lond. Math. Soc. 48:94–98, 2016). Our considerations are based on the properties of polars of plane singularities in characteristic p.

Evelia R. García Barroso, Arkadiusz Płoski

Foliations in the Plane Uniquely Determined by Minimal Subschemes of its Singularities

Abstract

Let \( \mathbb {P}^n \) be the projective space over an algebraically closed ground field K. In a previous paper, we have shown that the space of foliations by curves of degree greater than or equal to two which are uniquely determined by a subscheme of minimal degree of its scheme of singularities, contains a nonempty Zariski-open subset and hence, that the set of non-degenerate foliations with this property contains a Zariski-open subset. Moreover, we posed the question whether every non-degenerate foliation in \( \mathbb {P}^2 \) has this property. In this paper, we prove that this is true, in \( \mathbb {P}^2 \), in degrees 4 and 5.

Jorge Olivares

Newton Transformations and the Motivic Milnor Fiber of a Plane Curve

Abstract

In this article we give an expression of the motivic Milnor fiber at the origin of a polynomial in two variables with coefficients in an algebraically closed field. The expression is given in terms of some motives associated to the faces of the Newton polygons appearing in the Newton algorithm. In the complex setting, we deduce a computation of the Euler characteristic of the Milnor fiber in terms of the area of the surfaces under the Newton polygons encountered in the Newton algorithm which generalizes the Milnor number computation by Kouchnirenko in the isolated case.

Pierrette Cassou-Noguès, Michel Raibaut

Nash Modification on Toric Curves

Abstract

We revisit the problem of resolution of singularities of toric curves by iterating the Nash modification. We give a bound on the number of iterations required to obtain the resolution. We also introduce a different approach on counting iterations by dividing the combinatorial algorithm of the Nash modification of toric curves into several division algorithms.

Daniel Duarte, Daniel Green Tripp

A Recursive Formula for the Motivic Milnor Fiber of a Plane Curve

Abstract

We find a recursive formula for the motivic Milnor fiber of an irreducible plane curve, using the notions of a truncation and derived curve. We then apply natural transformations to obtain a similar recursion for the Hodge-theoretic spectrum.

Manuel González Villa, Gary Kennedy, Lee J. McEwan

On Hasse–Schmidt Derivations: The Action of Substitution Maps

Abstract

We study the action of substitution maps between power series rings as an additional algebraic structure on the groups of Hasse–Schmidt derivations. This structure appears as a counterpart of the module structure on classical derivations.

Luis Narváez Macarro

An Introduction to Resolution of Singularities via the Multiplicity

Abstract

In these notes we study properties of the multiplicity at points of a variety X over a perfect field. We focus on properties that can be studied using ramification method, such as discriminants and some generalized discriminants that we shall introduce. We also show how these methods lead to an alternative proof of resolution of singularities for varieties over fields of characteristic zero.

Diego Sulca, Orlando Villamayor U.

Platonic Surfaces

Abstract

We define the notion of Platonic surfaces. These are anticanonical smooth projective rational surfaces defined over any fixed algebraically closed field of arbitrary characteristic and having the projective plane as a minimal model with very nice geometric properties. We prove that their Cox rings are finitely generated. In particular, they are extremal and their effective monoids are finitely generated. Thus, these Platonic surfaces are built from points of the projective plane which are in good position. It is worth noting that not only their Picard number may be big but also an anticanonical divisor may have a very large number of irreducible components.

Brenda Leticia De La Rosa-Navarro, Gioia Failla, Juan Bosco Frías-Medina, Mustapha Lahyane, Rosanna Utano

Coverings of Rational Ruled Normal Surfaces

Abstract

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 \(\mathbb {Q}\)-normal crossing divisors.

Enrique Artal Bartolo, José Ignacio Cogolludo-Agustín, Jorge Martín-Morales

Ulrich Bundles on Veronese Surfaces

Abstract

It is a longstanding problem to determine whether the d-uple Veronese embedding of \({\mathbb {P}}^k\) supports a rank r Ulrich bundle. In this short note, we explicitly determine the integers d and r such that rank r Ulrich bundles on \({\mathbb {P}}^2\) for the Veronese embedding \({\mathcal {O}}(d)\) exist and, in particular, we solve Conjecture A.1 in Coskun and Genc (Proc Am Math Soc 145:4687–4701, 2017).

Laura Costa, Rosa Maria Miró-Roig

Multiple Structures on Smooth on Singular Varieties

Abstract

Let k an algebraically closed field, char k = 0. Let C be an irreducible nonsingular curve such that rC = S ∩ F, \(r\in \mathbb {N}\), where S and F are two surfaces in \(\mathbb {P}^3\) and all the singularities of F are of the form z ^p = x ^ps − y ^ps, p prime, \(s\in \mathbb {N} \). We prove that C can never pass through such kind of singularities of a surface, unless r = pa, \(a\in \mathbb {N}\). These singularities are Kodaira singularities.

M. R. Gonzalez-Dorrego

Smoothness in Some Varieties with Dihedral Symmetry and the DFT Matrix

Abstract

We study the smoothness question for some families of real and complex varieties with cyclic or dihedral symmetry. This question is related to deep properties of the Vandermonde matrix on the roots of unity, also known as the Discrete Fourier Transform matrix. We present some partial results on these questions.

Santiago López de Medrano

The Greedy Algorithm and the Cohen-Macaulay Property of Rings, Graphs and Toric Projective Curves

Abstract

It is shown in this paper how a solution for a combinatorial problem obtained from applying the greedy algorithm is guaranteed to be optimal for those instances of the problem that, under an appropriate algebraic representation, satisfy the Cohen-Macaulay property known for rings and modules in Commutative Algebra. The choice of representation for the instances of a given combinatorial problem is fundamental for recognizing the Cohen-Macaulay property. Departing from an exposition of the general framework of simplicial complexes and their associated Stanley-Reisner ideals, wherein the Cohen-Macaulay property is formally defined, a review of other equivalent frameworks more suitable for graphs or arithmetical problems will follow. In the case of graph problems a better framework to use is the edge ideal of Rafael Villarreal. For arithmetic problems it is appropriate to work within the semigroup viewpoint of toric geometry developed by Antonio Campillo and collaborators.

Argimiro Arratia

Binomial Ideals and Congruences on

Abstract

A congruence on \(\mathbb {N}^n\) is an equivalence relation on \(\mathbb {N}^n\) that is compatible with the additive structure. If \(\Bbbk \) is a field, and I is a binomial ideal in \(\Bbbk [X_1,\dots ,X_n]\) (that is, an ideal generated by polynomials with at most two terms), then I induces a congruence on \(\mathbb {N}^n\) by declaring u and v to be equivalent if there is a linear combination with nonzero coefficients of X ^u and X ^v that belongs to I. While every congruence on \(\mathbb {N}^n\) arises this way, this is not a one-to-one correspondence, as many binomial ideals may induce the same congruence. Nevertheless, the link between a binomial ideal and its corresponding congruence is strong, and one may think of congruences as the underlying combinatorial structures of binomial ideals. In the current literature, the theories of binomial ideals and congruences on \(\mathbb {N}^n\) are developed separately. The aim of this survey paper is to provide a detailed parallel exposition, that provides algebraic intuition for the combinatorial analysis of congruences. For the elaboration of this survey paper, we followed mainly (Kahle and Miller Algebra Number Theory 8(6):1297–1364, 2014) with an eye on Eisenbud and Sturmfels (Duke Math J 84(1):1–45, 1996) and Ojeda and Piedra Sánchez (J Symbolic Comput 30(4):383–400, 2000).

Laura Felicia Matusevich, Ignacio Ojeda

The K-Theory of Toric Schemes Over Regular Rings of Mixed Characteristic

Abstract

We show that if X is a toric scheme over a regular commutative ring k then the direct limit of the K-groups of X taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was previously known for regular commutative rings containing a field. The affine case of our result was conjectured by Gubeladze. We prove analogous results when k is replaced by an appropriate K-regular, not necessarily commutative k-algebra.

G. Cortiñas, C. Haesemeyer, M. E. Walker, C. A. Weibel

On Finite and Nonfinite Generation of Associated Graded Rings of Abhyankar Valuations

Abstract

We consider the condition that the associated graded ring of a local ring along an Abhyankar valuation is finitely generated. We characterize when this holds for regular local rings of domension two, and give some examples of maximal rank Abhyankar valuations for which the associated graded ring is not finitely generated on two dimensional normal local rings and regular local rings of dimension larger than two. We characterize two dimensional normal local rings for which all divisorial valuations which dominate the ring have a finitely generated associated graded ring.

Steven Dale Cutkosky

Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs

Abstract

In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertex-weighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equality between its ordinary and symbolic powers. We give an effective characterization of the Cohen–Macaulay vertex-weighted oriented forests. For edge ideals of transitive weighted oriented graphs we show that Alexander duality holds. It is shown that edge ideals of weighted acyclic tournaments are Cohen–Macaulay and satisfy Alexander duality.

Philippe Gimenez, José Martínez-Bernal, Aron Simis, Rafael H. Villarreal, Carlos E. Vivares

Asymptotics of Reduced Algebraic Curves Over Finite Fields

Abstract

The number A(q) shows the asymptotic behaviour of the quotient of the number of rational points over the genus of non-singular absolutely irreducible curves over \(\mathbb {F}_{q}\,\). Research on bounds for A(q) is closely connected with the so-called asymptotic main problem in Coding Theory. In this paper, we study some generalizations of this number for non-irreducible curves, their connection with A(q) and their application in Coding Theory. We also discuss the possibility of constructing codes from non-irreducible curves, both from theoretical and practical point of view.

J. I. Farrán

The Poincaré Polynomial of a Linear Code

Abstract

We introduce the Poincaré polynomial of a linear q-ary code and its relation to the corresponding weight enumerator. The question of whether the Poincaré polynomial is a complete invariant is answered affirmatively for q = 2, 3 and negatively for q ≥ 4. Finally we determine this polynomial for MDS codes and, by means of a recursive formula, for binary Reed-Muller codes.

Carlos Galindo, Fernando Hernando, Francisco Monserrat, Ruud Pellikaan

The Metric Structure of Linear Codes

Abstract

The bilinear form with associated identity matrix is used in coding theory to define the dual code of a linear code, also it endows linear codes with a metric space structure. This metric structure was studied for generalized toric codes and a characteristic decomposition was obtained, which led to several applications as the construction of stabilizer quantum codes and LCD codes. In this work, we use the study of bilinear forms over a finite field to give a decomposition of an arbitrary linear code similar to the one obtained for generalized toric codes. Such a decomposition, called the geometric decomposition of a linear code, can be obtained in a constructive way; it allows us to express easily the dual code of a linear code and provides a method to construct stabilizer quantum codes, LCD codes and in some cases, a method to estimate their minimum distance. The proofs for characteristic 2 are different, but they are developed in parallel.

Diego Ruano

On Some Properties of A Inherited by C b(X, A)

Abstract

Let X be a completely regular Hausdorff space or a pseudocompact Hausdorff space. We denote by C(X, A) the algebra of all continuous functions on X with values in a complex unital locally pseudo-convex algebra A. Let C _b(X, A) be its subalgebra consisting of all bounded continuous functions endowed with the topology given by the uniform pseudo-seminorms of A on X. In this paper we examine some properties of A that are inherited by C _b(X, A); these properties are projective limit decomposition, inversion, involution, spectral properties and metrizability.

Alejandra García, Lourdes Palacios, Carlos Signoret

A Fractional Partial Differential Equation for Theta Functions

Abstract

We find that theta functions are solutions of a fractional partial differential equation that generalizes the diffusion equation. This equation is the limit of a sequence of differential equations for the partial sums of theta functions where the fractional derivatives are given as differentiation matrices for trigonometric polynomials in their Fourier representation, i.e., given as similarities of diagonal matrices under the ordinary discrete Fourier transform. This fact enables the fast numerical computation of fractional partial derivatives of theta functions and elliptic integrals.

Rafael G. Campos

On Continued Fractions

Abstract

This paper on the geometry, algebra and arithmetics of continued fractions is based on a lecture for students, teachers and a non-specialist audience, beginning with the history of the golden number and Fibonacci sequence, continued fractions of rational and irrational numbers, Lagrange theorem on periodicity of continued fractions for quadratic irrationals, Klein’s geometric interpretation of the convergents as integer points, Jung-Hirzebruch continued fractions with negative signs and two dimensional singularities, higher dimensional generalizations, and ending with a result on a periodic generalized 3-dimensional continued fraction for a cubic irrational.

Gerardo Gonzalez Sprinberg