Mathematics - Algebraic Geometry Publications (50)


Mathematics - Algebraic Geometry Publications

The paper studies a relation between fundamental group of the complement to a plane singular curve and the orbifold pencils containing it. The main tool is the use of Albanese varieties of cyclic covers ramified along such curves. Our results give sufficient conditions for a plane singular curve to belong to an orbifold pencil, i. Read More

We relate the counting of honeycomb dimer configurations on the cylinder to the counting of certain vertices in Kirillov-Reshetikhin crystal graphs. We show that these dimer configurations yield the quantum Kostka numbers of the small quantum cohomology ring of the Grassmannian, i.e. Read More

Let $A$ be an abelian variety over the function field $K$ of a curve over a finite field. We provide several conditions ensuring that $A(K^{\rm perf})$ is finitely generated. This gives partial answers to questions of Scanlon and Ziegler on the one hand and Esnault and Langer on the other. Read More

We say that a formal power series $\sum a_n z^n$ with rational coefficients is a 2-function if the numerator of the fraction $a_{n/p}-p^2 a_n$ is divisible by $p^2$ for every prime number $p$. One can prove that 2-functions with rational coefficients appear as building block of BPS generating functions in topological string theory. Using the Frobenius map we define 2-functions with coefficients in algebraic number fields. Read More

In this paper, we study divisorial ideals of a Hibi ring which is a toric ring arising from a partially ordered set. We especially characterize the special class of divisorial ideals called conic using the associated partially ordered set. Using our observation of conic divisorial ideals, we also construct a module giving a non-commutative crepant resolution (= NCCR) of the Segre product of polynomial rings. Read More

Let $X$ be a connected scheme, smooth and separated over an algebraically closed field $k$ of characteristic $p\geq 0$, let $f:Y\rightarrow X$ be a smooth proper morphism and $x$ a geometric point on $X$. We prove that the tensor invariants of bounded length $\leq d$ of $\pi_1(X,x)$ acting on the \'etale cohomology groups $H^*(Y_x,F_\ell)$ are the reduction modulo-$\ell$ of those of $\pi_1(X,x)$ acting on $H^*(Y_x,Z_\ell)$ for $\ell$ greater than a constant depending only on $f:Y\rightarrow X$, $d$. We apply this result to show that the geometric variant with $F_\ell$-coefficients of the Grothendieck-Serre semisimplicity conjecture -- namely that $\pi_1(X,x)$ acts semisimply on $H^*(Y_x,F_\ell)$ for $\ell\gg 0$ -- is equivalent to the condition that the image of $\pi_1(X,x)$ acting on $H^*(Y_x,Q_\ell)$ is `almost maximal' (in a precise sense; what we call `almost hyperspecial') with respect to the group of $Q_\ell$-points of its Zariski closure. Read More

We show that limit linear series spaces for chains of curves are reduced. Using new advances in the foundations of limit linear series, we then use degenerations to study the question of connectedness for spaces of linear series with imposed ramification at up to two points. We find that in general, these spaces may not be connected even when they have positive dimension, but we prove a criterion for connectedness which generalizes the theorem previously proved by Fulton and Lazarsfeld in the case without imposed ramification. Read More

In this article, we introduce a notion of non-degeneracy, with respect to certain Newton polyhedra, for rational functions over non-Archimedean locals fields of arbitrary characteristic. We study the local zeta functions attached to non-degenerate rational functions, we show the existence of a meromorphic continuation for these zeta functions, as rational functions of $q^{-s}$, and give explicit formulas. In contrast with the classical local zeta functions, the meromorphic continuation of zeta functions for rational functions have poles with positive and negative real parts. Read More

Computations over the rational numbers often suffer from intermediate coefficient swell. One solution to this problem is to apply the given algorithm modulo a number of primes and then lift the modular results to the rationals. This method is guaranteed to work if we use a sufficiently large set of good primes. Read More

This note is based on the plenary talk given by the second author at MACIS 2015, the Sixth International Conference on Mathematical Aspects of Computer and Information Sciences. Motivated by some of the work done within the Priority Programme SPP 1489 of the German Research Council DFG, we discuss a number of current challenges in the development of Open Source computer algebra systems. The main focus is on algebraic geometry and the system Singular. Read More

These are expanded lecture notes from lectures given at the Workshop on higher structures at MATRIX Melbourne. These notes give an introduction to Feynman categories and their applications. Feynman categories give a universal categorical way to encode operations and relations. Read More

We use standard constructions in algebraic geometry and homological algebra to extend the decomposition and hard Lefschetz theorems of T. Mochizuki and C. Sabbah so that they remains valid without the quasi-projectivity assumptions. Read More

Recently, the singular support and the characteristic cycle of an \'etale sheaf on a smooth variety over a perfect field are constructed by Beilinson and Saito, respectively. In this article, we extend the singular support to a relative situation. As an application, we prove the generic constancy for singular supports and characteristic cycles of \'etale sheaves on a smooth fibration. Read More

We address the question about the birational geometry of the strata of holomorphic differentials. Using pencils on K3 surfaces in small genera we can construct rational curves on strata of holomorphic differentials and prove uniruledness in several cases. Unirationality is proven for some partition range in small genera. Read More

Assume that $M(\mathcal{T})$ is a rational homology sphere plumbed 3-manifold associated with a connected negative definite graph $\mathcal{T}$. We consider the combinatorial multivariable Poincar\'e series associated with $\mathcal{T}$ and its counting functions, which encode rich topological information. Using the `periodic constant' of the series (with reduced variables) we prove surgery formulae for the normalized Seiberg-Witten invariants: the periodic constant appears as the difference of the Seiberg-Witten invariants associated with $M(\mathcal{T})$ and $M(\mathcal{T}\setminus\mathcal{I})$, where $\mathcal{I}$ is an arbitrary subset of the set of vertices of $\mathcal{T}$. Read More

We compute Betti numbers of both the components of the moduli space of rank 2 semi-stable torsion-free sheaves with fixed determinant over a reducible nodal curve with two smooth components intersecting at a node. We also compute the intersection Betti numbers of the moduli space. Read More

These notes are based on a sequence of five lectures given to graduate students. The main goal is to prove the so-called Painleve property for semi-simple Frobenius manifolds. Read More

In this paper, we define and construct canonical filtered $F$-crystals with $G$-structure over the integral models for Shimura varieties of abelian type at hyperspecial level defined by Kisin. We check that these are related by $p$-adic comparison theorems to the usual lisse sheaves, and as an application we also use this to show that the Galois representations generated from the $p$-adic \'etale cohomology of Shimura varieties with nontrivial coefficient sheaves are crystalline, at least in the case of proper abelian type Shimura varieties. Read More

We study the irreducible components of the moduli space of instanton sheaves on $\mathbb{P}^3$, that is rank 2 torsion free sheaves $E$ with $c_1(E)=c_3(E)=0$ satisfying $h^1(E(-2))=h^2(E(-2))=0$. In particular, we classify all instanton sheaves with $c_2(E)\le4$, describing all the irreducible components of their moduli space. A key ingredient for our argument is the study of the moduli space ${\mathcal T}(d)$ of stable sheaves on $\mathbb{P}^3$ with Hilbert polynomial $P(t)=d\cdot t$, which contains, as an open subset, the moduli space of rank 0 instanton sheaves of multiplicity $d$; we describe all the irreducible components of ${\mathcal T}(d)$ for $d\le4$. Read More

We present a new family of monads whose cohomology is a stable rank two vector bundle on $\mathbb{P}^3$. We also study the irreducibility and smoothness together with a geometrical description of some of these families. Such facts are used to prove that the moduli space of stable rank two vector bundles with trivial determinant and second Chern class equal to 5 has exactly three irreducible components. Read More

Motivated by questions in real enumerative geometry we investigate the problem of the number of flats simultaneously tangent to several convex hypersurfaces in real projective space from a probabilistic point of view. More precisely, we say that smooth convex hypersurfaces $X_1, \ldots, X_{d_{k,n}}\subset \mathbb{R}\textrm{P}^n$, where $d_{k,n}=(k+1)(n-k)$, are in random position if each one of them is randomly translated by elements $g_1, \ldots, g_{{d_{k,n}}}$ sampled independently from the Orthogonal group with the uniform distribution; we denote by $\tau_k(X_1, \ldots, X_{d_{k,n}})$ the average number of $k$-dimensional projective subspaces (flats) which are simultaneously tangent to all the hypersurfaces. We prove that $$ \tau_k(X_1, \ldots, X_{d_{k,n}})={\delta}_{k,n} \cdot \prod_{i=1}^{d_{k,n}}\frac{|\Omega_k(X_i)|}{|\textrm{Sch}(k,n)|},$$ where ${\delta}_{k,n}$ is the expected degree (the average number of $k$-flats incident to $d_{k,n}$ many random $(n-k-1)$-flats), $|\textrm{Sch}(k,n)|$ is the volume of the Special Schubert variety of $k$-flats meeting a $(n-k-1)$-flat and $|\Omega_k(X)|$ is the volume of the manifold of all $k$-flats tangent to $X$. Read More

We consider the closed locus of $r$-tuples of hypersurfaces in $\mathbb{P}^r$ with positive dimensional intersection, and show in a large range of degrees that its largest component is the locus of $r$-tuples of hypersurfaces whose intersection contains a line. We then apply our methods to obtain new results on the largest components of the locus of hypersurfaces with positive dimensional singular loci and the locus of smooth hypersurfaces with a higher dimensional family of lines through a point than expected. Read More

We revisit some of the basic results of generic vanishing theory, as pioneered by Green and Lazarsfeld, in the context of constructible sheaves. Using the language of perverse sheaves, we give new proofs of some of the basic results of this theory. Our approach is topological/arithmetic, and avoids Hodge theory. Read More

We give a framework to produce constructible functions from natural functors between categories, without need of a morphism of moduli spaces to model the functor. We show using the Riemann-Hilbert correspondence that any natural (derived) functor on constructible sheaves on smooth complex algebraic varieties can be used to construct a special kind of constructible sets, called absolute sets, generalizing a notion introduced by Simpson in presence of moduli. We conjecture that the absolute sets of local systems satisfy a "special varieties package", among which is an analog of the Manin-Mumford, Mordell-Lang, and Andr\'e-Oort conjectures. Read More

Let $X$ be a smooth complex projective variety. In 2002, Bridgeland defined a notion of stability for the objects in $D^b(X)$, the bounded derived category of coherent sheaves on $X$, which generalized the notion of slope stability for vector bundles on curves. There are many nice connections between stability conditions on $X$ and the geometry of the variety. Read More

We present an algorithm for explicitly computing the number of generators of the stable cohomology algebra of any rationally smooth partial toroidal compactification of ${\mathcal A}_g$, satisfying certain additivity and finiteness properties, in terms of the combinatorics of the corresponding toric fans. In particular the algorithm determines the stable cohomology of the matroidal partial compactification, in terms of simple regular matroids that are irreducible with respect to the 1-sum operation, and their automorphism groups. The algorithm also applies to compute the stable Betti numbers in close to top degree for the perfect cone toroidal compactification. Read More

This paper is devoted to study multiplicity and regularity as well as to present some classifications of complex analytic sets. We present an equivalence for complex analytical sets, namely blow-spherical equivalence and we receive several applications with this new approach. For example, we reduce to homogeneous complex algebraic sets a version of Zariski's multiplicity conjecture in the case of blow-spherical homeomorphism, we give some partial answers to the Zariski's multiplicity conjecture, we show that a blow-spherical regular complex analytic set is smooth and we give a complete classification of complex analytic curves. Read More

Algebraic surfaces of general type with $q=0$, $p_g=2$ and $K^2=1$ have been studied by Horikawa. In this paper we consider a $16$-dimensional family of special Horikawa surfaces which are certain bidouble covers of $\mathbb{P}^2$. The construction is motivated by that of special Kunev surfaces which are counterexamples for infinitesimal Torelli and generic global Torelli problem. Read More

We consider the population of critical points generated from the critical point of the master function with no variables, which is associated with the trivial representation of the twisted affine Lie algebra $A^{(2)}_{2n}$. The population is naturally partitioned into an infinite collection of complex cells $\mathbb{C}^m$, where $m$ are some positive integers. For each cell we define an injective rational map $\mathbb{C}^m \to M(A^{(2)}_{2n})$ of the cell to the space $M(A^{(2)}_{2n})$ of Miura opers of type $A^{(2)}_{2n}$. Read More

Singular limits of 6D F-theory compactifications are often captured by T-branes, namely a non-abelian configuration of intersecting 7-branes with a nilpotent matrix of normal deformations. The long distance approximation of such 7-branes is a Hitchin-like system in which simple and irregular poles emerge at marked points of the geometry. When multiple matter fields localize at the same point in the geometry, the associated Higgs field can exhibit irregular behavior, namely poles of order greater than one. Read More

This is the third paper in a series of work on weighted stable elliptic surfaces - elliptic fibrations with section and marked fibers weighted between zero and one. Motivated by Hassett's weighted pointed stable curves, we use the log minimal model program to construct compact moduli spaces parameterizing these objects. Moreoever, we show that the domain of weights admits a "wall and chamber" structure, we describe the induced wall crossing morphisms on the moduli spaces as the weight vector varies, and describe the surfaces that appear on the boundary of the moduli space. Read More

We study the fundamental relationship between stable quotient invariants and the B-model for local CP2 in all genera. Our main result is a direct geometric proof of the holomorphic anomaly equation in the precise form predicted by B-model physics. The method yields new holomorphic anomaly equations for an infinite class of twisted theories on projective spaces. Read More

The purpose of this article is to clarify the question what makes motives $\mathbb{A}^1$-homotopy invariance. we give construction of the stable model category of nilpotent invariant motives $\mathcal{M}ot_{\operatorname{dg}}^{\operatorname{nilp}}$ and define the nilpotent invriant motives associated with schemes and relative exact categories. For a noetherian scheme $X$, there are two kind of motives associated with $X$ in the homotopy category $\operatorname{Ho}(\mathcal{M}ot^{\operatorname{nilp}}_{\operatorname{dg}})$, namely $M_{\operatorname{nilp}}(X)$ and $M_{\operatorname{nilp}}'(X)$. Read More

We study the $m$-th Gauss map in the sense of F.~L.~Zak of a projective variety $X \subset \mathbb{P}^N$ over an algebraically closed field in any characteristic. Read More

We present several upper bounds for the height of global residues of rational forms on an affine variety. As a consequence, we deduce upper bounds for the height of the coefficients in the Bergman-Weil trace formula. We also present upper bounds for the degree and the height of the polynomials in the elimination theorem on an affine variety. Read More

Long linear codes constructed from toric varieties over finite fields, their multiplicative structure and decoding. The main theme is the inherent multiplicative structure on toric codes. The multiplicative structure allows for \emph{decoding}, resembling the decoding of Reed-Solomon codes and aligns with decoding by error correcting pairs. Read More

In this article we study Brill-Noether loci of moduli space of stable bundles over smooth surfaces. We define Petri map as an analogy with the case of curves. We show the non-emptiness of certain Brill-Noether loci over very general quintic hypersurface in $\mathbb{P}^3$, and use the Petri map to produce components of expected dimension. Read More

To a univariate monic polynomial is attached a special planar forest that is called the picture of the polynomial. Isotopy classes of pictures are called signatures. All combinatorially possible signatures are realized and spaces of polynomials realizing a given signature are contractible. Read More

In algebraic geometry there is the notion of a height pairing of algebraic cycles, which lies at the confluence of arithmetic, Hodge theory and topology. After explaining a motivating example situation, we introduce new directions in this subject. Read More

On arithmetic surfaces over local rings of integers we examine whether any geometric point has a basis of \'etale neighborhoods whose $\mathfrak{c}$-completed \'etale homotopy types are of type $K(\pi,1)$ for a given full class~$\mathfrak{c}$ of finite groups. Read More

We study the germs at the origin of $G$-representation varieties and the degree 1 cohomology jump loci of fundamental groups of quasi-projective manifolds. Using the Morgan-Dupont model associated to a convenient compactification of such a manifold, we relate these germs to those of their infinitesimal counterparts, defined in terms of flat connections on those models. When the linear algebraic group $G$ is either $\textrm{SL}_2(\mathbb{C})$ or its standard Borel subgroup and the depth of the jump locus is 1, this dictionary works perfectly, allowing us to describe in this way explicit irreducible decompositions for the germs of these embedded jump loci. Read More

We show that a reduct of the Zariski structure of an algebraic curve which is not locally modular interprets a field, answering a question of Zilber's. Read More

In this expository note, we review the standard formulation of mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, and compare this construction to a description of mirror symmetry for K3 surfaces which relies on a sublattice of the Picard lattice. We then show how to combine information about the Picard group of a toric ambient space with data about automorphisms of the toric variety to identify families of K3 surfaces with high Picard rank. Read More

We present new results on equisingularity and equinormalizability of families with isolated non-normal singularities (INNS) of arbitrary dimension. We define a $\delta$-invariant and a $\mu$-invariant for an INNS and prove necessary and sufficient numerical conditions for equinormalizability and weak equinormalizability using $\delta$ and $\mu$. Moreover, we determine the number of connected components of the Milnor fibre of an arbitrary INNS. Read More

We analyze the modular properties of D3-brane instanton corrections to the hypermultiplet moduli space in type IIB string theory compactified on a Calabi-Yau threefold. In Part I, we found a necessary condition for the existence of an isometric action of S-duality on this moduli space: the generating function of DT invariants in the large volume attractor chamber must be a vector-valued mock modular form with specified modular properties. In this work, we prove that this condition is also sufficient at two-instanton order. Read More

We study the four-dimensional low energy effective $\mathcal{N}=1$ supergravity theory of the dimensional reduction of M-theory on $G_2$-manifolds, which are constructed by Kovalev's twisted connected sum gluing suitable pairs of asymptotically cylindrical Calabi--Yau threefolds $X_{L/R}$ augmented with a circle $S^1$. In the Kovalev limit the Ricci-flat $G_2$-metrics are approximated by the Ricci-flat metrics on $X_{L/R}$ and we identify the universal modulus --- the Kovalevton --- that parametrizes this limit. We observe that the low energy effective theory exhibits in this limit gauge theory sectors with extended supersymmetry. Read More

We compute toric degenerations arising from the tropicalization of the full flag varieties $\mathcal{F}\ell_4$ and $\mathcal{F}\ell_5$ embedded in a product of Grassmannians. For $\mathcal{F}\ell_4$ and $\mathcal{F}\ell_5$ we compare toric degenerations arising from string polytopes and the FFLV polytope with those obtained from the tropicalization of the flag varieties. We also present a general procedure to find toric degenerations in the cases where the initial ideal arising from a cone of the tropicalization of a variety is not prime. Read More

In this paper we enumerate nonhyperelliptic superspecial curves of genus $4$ over prime fields of characteristic $p\le 11$. Our algorithm works for nonhyperelliptic curves over an arbitrary finite field in characteristic $p \ge 5$. We execute the algorithm for prime fields of $p\le 11$ with our implementation on a computer algebra system Magma. Read More

This is a survey article about parabolic bundles and parabolic Higgs bundles. Read More

This is a survey article on recent results on vector bundles on symmetric product of non-singular projective curves. Read More