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In this paper, we study the birational geometry of the Quot schemes of trivial bundles on $\mathbb{P}^1$ by constructing small $\mathbb{Q}$-factorial modifications of the Quot schemes as suitable moduli spaces. We determine all the models which appear in the minimal model program on the Quot schemes. |
Oort-Zink proved that a $p$-divisible group over a normal base in characteristic $p$ with constant Newton polygon is isogenous to a $p$-divisible group admitting a slope filtration. In this paper, we generalize this result to log $p$-divisible groups. |
We prove generalized Virasoro constraints for the relative Gromov-Witten theories of all nonsingular target curves. Descendents of the even cohomology classes are studied first by localization, degeneration, and completed cycle methods. |
We provide a complete motivic decomposition of a twisted form of a smooth hyperplane section of Gr(3,6). This variety is a norm variety corresponding to a (3,3)-symbol. |
In the present note we study absolute linear Harbourne constants. These are invariants which were introduced in order to relate the lower bounds on the selfintersection of negative curves on birationally equivalent surfaces to the complexity of the birational map between them. |
We study the geometry and cohomology of algebraic super curves, using a new contour integral for holomorphic differentials. For a class of super curves (``generic SKP curves'') we define a period matrix. |
We prove that, under certain conditions, the existence of a curve of $(m+2)$-secants to the Kummer variety of an indecomposable principally polarized abelian variety $X$, represents $m$-times the minimal cohomological class in $X$. In the case of $m=2$, we find an involution of such curve. |
The notion of a Joyce structure was introduced in <a href="https://arxiv.org/abs/1912.06504" data-arxiv-id="1912.06504" class="link-https">arXiv:1912.06504</a> to describe the geometric structure on the space of stability conditions of a CY3 category naturally encoded by the Donaldson-Thomas invariants. In this paper ... |
Motivated by a conjecture of Xiao, we study families of coverings of elliptic curves and their corresponding Prym map $\Phi$. More precisely, we describe the codifferential of the period map $P$ associated to $\Phi$ in terms of the residue of meromorphic $1$-forms and then we use it to give a characterization for the ... |
Synchronization in networks of interconnected oscillators is a fascinating phenomenon that appear naturally in many independent fields of science and engineering. A substantial amount of work has been devoted to understanding all possible synchronization configurations on a given network. |
In previous work, we constructed for a smooth complex variety $X$ and for a linear algebraic group $G$ a mixed Hodge structure on the complete local ring $\widehat{\mathcal{O}}_\rho$ to the moduli space of representations of the fundamental group $\pi_1(X,x)$ into $G$ at a representation $\rho$ underlying a variation o... |
Let $X$ be a smooth projective variety. We construct partial Okounkov bodies associated to Hermitian pseudo-effective line bundles $(L,\phi)$ on $X$. |
Let A be a finitely generated associative algebra over an algebraically closed field. We characterize the finite dimensional modules over A whose orbit closures are regular varieties. |
Numerical algebraic geometry has a close relationship to intersection theory from algebraic geometry. We deepen this relationship, explaining how rational or algebraic equivalence gives a homotopy. |
The aim of this paper is to give a unified definition of a large class of discriminants arising in algebraic geometry using the discriminant of a morphism of locally free sheaves. The discriminant of a morphism of locally free sheaves has a geometric definition in terms of grassmannian bundles, tautological sequences ... |
This article studies descent theory in the setting of Berkovich spaces. We give sufficient conditions for a given fibered category over the category of k-affinoid algebras to be a stack for the Berkovich analogue of the faithfully-flat topology. |
We give an overview of the category of subgroups of the modular group, incorporating both the tame part, i.e. finite index subgroups, and the non-tame part, i.e. the rest. We also discuss arithmetic related questions which exist in both the tame part (via Belyi's theorem) and the non-tame part. |
We introduce in this paper a hypercohomology version of the resonance varieties and obtain some relations to the characteristic varieties of rank one local systems on a smooth quasi-projective complex variety $M$, see Theorem (3.1) and Corollaries (3.2) and (4.2). A logarithmic resonance variety is also considered in ... |
Let X_o be a complex weighted-homogeneous complete intersection germ, (possibly non-reduced). Let X be a perturbation of X_o by ``higher-order-terms". |
Let J be the Jacobian of a smooth curve C of genus g, and let A(J) be the ring of algebraic cycles modulo algebraic equivalence on J, tensored with Q. We study in this paper the smallest Q-vector subspace R of A(J) which contains C and is stable under the natural operations of A(J) : intersection and Pontryagin product... |
We consider Calabi-Yau threefolds $X$ over an algebraically closed field $k$ of characteristic $p>0$ that are not liftable to characteristic $0$ or liftable ones with $p=2$. It is unknown whether Kodaira vanishing holds for these varieties. |
We consider the Quot scheme, R_{d}, compactifying the space of degree d maps from the projective line to the Grassmannian of lines. We give an algorithm for computing the degree of R_{d} under a "generalized Plücker embedding", this is a certain Gromov-Witten invariant. |
This is a survey paper: we discuss certain recent results, with some improvements. It will appear in the S. Cruz proceedings. |
The interpretation, due to T. Mabuchi, of the classical Futaki invariant of Fano toric manifolds is extended to the case of the Generalized Futaki invariant, introduced by W. Ding and G. Tian, of almost Fano toric varieties. As an application it is shown that the real part of the Generalized Futaki invariant is positi... |
For a real number $0<\epsilon<1/3$, we show that the anti-canonical volume of an $\epsilon$-klt Fano $3$-fold is at most $3200/\epsilon^4$ and the order $O(1/\epsilon^4)$ is sharp. |
We introduce multi-uniformized stacks as a generalization of the Abramovich--Hassett construction of uniformized twisted varieties. We prove an equivalence between the category of multi $\mathbb{Q}$-line bundles satisfying an analogue of Kollár's condition and the category of multi-uniformized twisted varieties, a... |
Let $X$ be a complete symmetric variety i.e. the wonderful compactification of a symmetric $G-$homogeneous space (where $G$ is a simply-connected semi-simple linear algebraic group). If $L$ is a line bundle over $X$ and if $C$ is a Bialynicki-Birula cell of codimension $c$ in $X$, then the Lie algebra $\mathfrak g$ of... |
Let $\Delta$ be a $3$-dimensional lattice polytope containing exactly one interior lattice point. We give a simple combinatorial formula for computing the stringy $E$-function of the $3$-dimensional canonical toric Fano variety $X_{\Delta}$ associated with the polytope $\Delta$. |
We consider the rational dynamical quantum group $E_y(gl_2)$ and introduce an $E_y(gl_2)$-module structure on $\oplus_{k=0}^n H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$, where $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$ is the equivariant cohomology algebra $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))$ of the cotangent bun... |
The $j$-invariant of a cubic curve is an isomorphism invariant parameterized by the moduli space of elliptic curves. The Hesse derivative of a curve $V(f)$ given by the homogeneous polynomial $f$ is $V(\mathcal{H}(f))$ where $\mathcal{H}(f)$ is a the determinant of the Hesse matrix of $f$. |
We establish a local model for the moduli space of holomorphic symplectic structures with logarithmic poles, near the locus of structures whose polar divisor is normal crossings. In contrast to the case without poles, the moduli space is singular: when the cohomology class of a symplectic structure satisfies certain l... |
Given a birational modification $X \to Y$ of complex projective varieties with fiber dimension 1 and rational singularities, consider the main component of Bridgeland's moduli space $W \to Y$ of perverse point sheaves on $X/Y$. We give criteria for the normalization of $W$ to coincide with the transform (flip/flop... |
We review the theory of Lagrangian fibrations of hyperkähler manifolds as initiated by Matsushita. We also discuss more recent work of Shen-Yin and Harder-Li-Shen-Yin. |
In this paper we study the Oort conjecture on Shimura subvarieties contained generically in the Torelli locus in the Siegel modular variety $\mathcal{A}_g$. Using the poly-stability of Higgs bundles on curves and the slope inequality of Xiao on fibred surfaces, we show that a Shimura curve $C$ is not contained generic... |
For any field k of characteristic at most 5 we exhibit an explicit smooth quartic surface in projective threespace over k with trivial automorphism group over the algebraic closure of k. We also show how this can be extended to higher characteristics. |
We prove that elliptic K3 surfaces over a number field which admit a second elliptic fibration satisfy the potential Hilbert property. Equivalently, the set of their rational points is not thin after a finite extension of the base field. |
A complete intersection $f_1=\cdots=f_k=0$ is schön, if $f_1=\cdots=f_j=0$ defines a schön subvariety of an algebraic torus for every $j\leqslant k$. This class includes nondegenerate complete intersections, critical loci of their coordinate projections, other simplest Thom--Boardman and multiple point strata of such ... |
We discuss some features of the so-called Zariski's multiplicity problem especially the application of the work of A'Campo on the zeta function of a monodromy of an isolated singularity of a complex hypersurface to the problem. |
We study the normal bundles of the exceptional sets of isolated simple small singularities in the higher dimension when the Picard group of the exceptional set is $\mathbb{Z}$ and the normal bundle of it has some good filtration. In particular, for the exceptional set is a projective space with the split normal bundle... |
Using the path lattice cohomology we provide a conceptual topological characterization of the geometric genus for certain complex normal surface singularities with rational homology sphere links, which is uniformly valid for all superisolated and Newton non--degenerate hypersurface singularities. |
We show the semi-continuity property of minimal log discrepancies for varieties which have a crepant resolution in the category of Deligne-Mumford stacks. Using this property, we also prove the ideal-adic semi-continuity problem for toric pairs. |
The aim of this note is to treat one distinguished example of a Calabi--Yau variety that appears as a small resolution of a Picard modular variety |
Let $G$ be a group and let $k$ be a field. Kaplansky's direct finiteness conjecture states that every one-sided unit of the group ring $k[G]$ must be a two-sided unit. |
Let $X$ be a closed subscheme embedded in a scheme $W$ smooth over a field ${\bf k}$ of characteristic zero, and let ${\mathcal I}(X)$ be the sheaf of ideals defining $X$. Assume that the set of regular points of $X$ is dense in $X$. |
In this paper a construction of quantum codes from self-orthogonal algebraic geometry codes is provided. Our method is based on the CSS construction as well as on some peculiar properties of the underlying algebraic curves, named Swiss curves. |
A plane cubic curve, defined over a field with valuation, is in honeycomb form if its tropicalization exhibits the standard hexagonal cycle. We explicitly compute such representations from a given j-invariant with negative valuation, we give an analytic characterization of elliptic curves in honeycomb form, and we off... |
In this paper, we explore a notion of nonabelian Hodge structure on the fundamental group of an algebraic variety. This is approach is compared to some alternative approaches due to Morgan, Hain and others. |
For certain complex projective manifolds (such as K3 surfaces and their higher dimensional analogues, the complex symplectic projective manifolds) the period map takes values in a locally symmetric variety of type IV. It is often an open embedding and in such cases it has been observed that the image is the complement... |
We define a filtration on the Chow groups of a smooth projective variety X over a field k by using the cycle map into continuous l-adic etale cohomology. The main theorem says that if k is a function field in one variable over a finite field, this filtration for zero cycles is of length at most one modulo the kernel o... |
We characterize the ideals $I$ of $\mathcal O_n$ of finite colength whose integral closure is equal to the integral closure of an ideal generated by pure monomials. This characterization, which is motivated by an inequality proven by Demailly and Pham, is given in terms of the log canonical threshold of $I$ and the se... |
We formulate Virasoro constraints for the generating functions of the intersection numbers on Hassett's moduli of weighted pointed curves and show that they are governed by the KdV integrable hierarchy. |
For a given triangle $\triangle ABC$, we define two sequences of points on line $BC$ and provide their generalizations to real functions such that centers of circumscribed circles around $A$ and adjacent points in subsequences generate a pencil of conics touching perpendicular bisectors of $AB$ and $AC$. |
Let X be an irreducible symplectic variety defined over a number field K. Assume either that X has Picard number at least two or that X has even second Betti number. We prove that there exist a finite algebraic field extension L/K and a density 1 set S of non-archimedean places of L such that the reduction of X at any... |
A family of curves over a discrete valuation ring is called semi-factorial if every line bundle on the generic fibre extends to a line bundle on the total space. In the nodal case, we give a sufficient and necessary condition for semi-factoriality, in terms of combinatorics of the dual graph of the special fibre. |
Given a strict simple degeneration $f \colon X\to C$ the first three authors previously constructed a degeneration $I^n_{X/C} \to C$ of the relative degree $n$ Hilbert scheme of $0$-dimensional subschemes. In this paper we investigate the geometry of this degeneration, in particular when the fibre dimension of $f$ is ... |
In this paper we focus on the problem of computing the number of moduli of the so called Severi varieties (denoted by V(|D|, \delta)), which parametrize universal families of irreducible, \delta-nodal curves in a complete linear system |D|, on a smooth projective surface S of general type. We determine geometrical and... |
We show that a central linear mapping of a projectively embedded Euclidean $n$-space onto a projectively embedded Euclidean $m$-space is decomposable into a central projection followed by a similarity if, and only if, the least singular value of a certain matrix has multiplicity $\ge 2m-n+1$. This matrix is arising, b... |
In a previous paper we introduced the notion of a D-Lie algebra $\tilde{L}$. A D-Lie algebra $\tilde{L}$ is an $A/k$-Lie-Rinehart algebra with a right $A$-module structure and a canonical central element $D$ satisfying several conditions. |
Saito gave a nice and efficient criterion to determine whether the module of logarithmic derivation associated with a reduced divisor in a complex variety is free or not. The aim of this note is to propose a new proof of this criterion, in the affine space, in the projective space, and for multiderivations, based on s... |
Let $p$ be a prime integer and $F$ a field of characteristic 0. Let $X$ be the {\em norm variety} of a symbol in the Galois cohomology group $H^{n+1}(F,\mu_p^{\otimes n})$ (for some $n\geq1$), constructed in the proof of the Bloch-Kato conjecture. |
Given a split reductive Chevalley group scheme G over Z and a parabolic subgroup scheme P in G, this paper constructs G-linear semiorthogonal decompositions of the bounded derived category of noetherian representations of P with each semiorthogonal component being equivalent to the bounded derived category of noetheria... |
We prove a few splitting criteria for vector bundles on a quadric hypersurface and Grassmannians. We give also some cohomological splitting conditions for rank 2 bundles on multiprojective spaces. |
In tropical geometry, given a curve in a toric variety, one defines a corresponding graph embedded in Euclidean space. We study the problem of reversing this process for curves of genus zero and one. |
The pure braid group \Gamma of a quadruply-punctured Riemann sphere acts on the SL(2,C)-moduli M of the representation variety of such sphere. The points in M are classified into \Gamma-orbits. |
There are two well known tasks, related to Newton polyhedra: to study invariants of singularities in terms of their Newton polyhedra, and to describe Newton polyhedra of resultants and discriminants. We introduce so called resultantal singularities, whose study in terms of Newton polyhedra unifies these two tasks to a... |
This paper is an elementary introduction to the theory of moduli spaces of curves and maps. As an application to enumerative geometry, we show how to count the number of bitangent lines to a projective plane curve of degree $d$ by doing intersection theory on moduli spaces. |
This paper proves a linear algebra result that has to do with the geometry of "widgets". For us a widget is a collection of n pairs of points in a vector space. |
This review is an elaboration of a presentation given at the Real algebraic geometry and singularities conference in honor of Wojciech Kucharz's 70th birthday in Krakow in 2022. |
We generalize Luna's fundamental lemma to smooth morphisms between stacks with good moduli spaces. We also give a precise condition for when it holds for non-smooth morphisms and versions for coherent sheaves and complexes. |
We construct and study a candidate for the standard motivic t-structure on the triangulated category of relative cohomological 1-motives with rational coefficients over a noetherian finite dimensional scheme S. This t-structure is defined as a generated t-structure, and we show it is non-degenerate. We relate its hear... |
Let ${\rm F}$ be a rank-2 semi-stable sheaf on the projective plane, with Chern classes $c_{1}=0,c_{2}=n$. The curve $\beta_{\rm F}$ of jumping lines of ${\rm F}$, in the dual projective plane, has degree $n$. |
We prove that two general Enriques surfaces defined over an algebraically closed field of characteristic different from $2$ are isomorphic if their Kuznetsov components are equivalent. We apply the same techniques to give a new simple proof of a conjecture by Ingalls and Kuznetsov relating the derived categories of th... |
Affine Deligne-Lusztig varieties are closely related to the special fibre of Newton strata in the reduction of Shimura varieties or of moduli spaces of $G$-shtukas. In almost all cases, they are not quasi-compact. |
In this paper we prove first a general theorem on semiorthogonal decompositions in derived categories of coherent sheaves for flat families over a smooth base. Based on the results of <a href="https://arxiv.org/abs/math.AG/0510670" data-arxiv-id="math.AG/0510670" class="link-https">math.AG/0510670</a>, we then show th... |
We prove an analogue of the Lindemann-Weierstrass theorem (that the exponentials of Q-linearly independent algebraic numbers are algebraically independent) for commutative algebraic groups G without unipotent quotients, over function fields. We concentrate on solutions to the the differential algebraic relations satis... |
In this paper we present a constructive method to characterize ideals of the local ring $\mathscr{O}_{\mathbb{C}^n,0}$ of germs of holomorphic functions at $0\in\mathbb{C}^n$ which arise as the moduli ideal $\langle f,\mathfrak{m}\, j(f)\rangle$, for some $f\in\mathfrak{m}\subset\mathscr{O}_{\mathbb{C}^n,0}$. A conseq... |
Given a closed subvariety X in a projective space, the rank with respect to X of a point p in this projective space is the least integer r such that p lies in the linear span of some r points of X. Let W_k be the closure of the set of points of rank with respect to X equal to k. For small values of k such loci are cal... |
We introduce a characteristic vector with respect to a regular triangulation of the momentum polytope to compute the Hurwitz polytope of a given smooth toric variety. As a result, we prove that the convex hull of such vectors of all regular triangulations is included in the Hurwitz polytope of a smooth toric surface. |
We study the geometry of D-bundles--locally projective D-modules--on algebraic curves, and apply them to the study of integrable hierarchies, specifically the multicomponent Kadomtsev-Petviashvili (KP) and spin Calogero-Moser (CM) hierarchies. We show that KP hierarchies have a geometric description as flows on moduli... |
Let X be an irreducible complex variety, S a stratification of X and F a holomorphic vector bundle on the open statum. We give geometric conditions on S and F that produce a natural extension of the k-th Chern class F as a class in the complex cohomology of X of Hodge level at least k. |
Let $\Bbbk$ be any field of characteristic zero, $X$ be a del Pezzo surface of degree~$2$ and $G$ be a group acting on $X$. In this paper we study $\Bbbk$-rationality questions for the quotient surface $X / G$. |
In this paper we give an example of a triangulated category, linear over a field of characteristic zero, which does not carry a DG-enhancement. The only previous examples of triangulated categories without a model have been constructed by Muro, Schwede and Strickland. |
In this paper, the containment problem for the defining ideal of a special type of zero dimensional subschemes of $\mathbb{P}^2$, so called quasi star configurations, is investigated. Some sharp bounds for the resurgence of these types of ideals are given. |
Let $C$ be a comb-like curve over $\mathbb{C}$, and $E$ be a vector bundle of rank $n$ on $C$. In this paper, we investigate the criteria for the semistability of the restriction of $E$ onto the components of $C$ when $E$ is given to be semistable with respect to a polarization $w$. |
The Fukaya category of a punctured surface can be reconstructed from a pair-of-pants decomposition using a formal construction that attaches a category to a trivalent graph. We extend this formal construction to include a choice of line field on the surface, this requires a certain decoration on the graph. |
We exhibit examples of slope-stable and modular vector bundles on a hyperkähler manifold of K3$^{[2]}$-type which move in a 20-dimensional family and study their algebraic properties. These are obtained by performing standard linear algebra constructions on the examples studied by O'Grady of (rigid) modular bundle... |
Cylindrical algebraic decomposition is a classical construction in real algebraic geometry. Although there are many algorithms to compute a cylindrical algebraic decomposition, their practical performance is still very limited. |
We consider Łojasiewicz inequalities for a non-degenerate holomorphic function with an isolated singularity at the origin. We give an explicit estimation of the Łojasiewicz exponent in a slightly weaker form than the assertion in <a href="http://Fukui.For" rel="external noopener nofollow" class="link-external link-htt... |
We study Coble surfaces in characteristic 2, in particular, singularities of their canonical coverings. As an application we classify Coble surfaces with finite automorphism group in characteristic 2. |
We consider a mixed function of type $H(\mathbf z,\bar {\mathbf z})=f(\mathbf z)\bar g(\mathbf z)$ where $f$ and $g$ are convenient holomorphic functions which have isolated critical points at the origin and we assume that the intersection $f=g=0$ is a complete intersection variety with an isolated singlarity at theori... |
We study the moduli of trigonal curves. We establish the exact upper bound of ${36(g+1)}/(5g+1)$ for the slope of trigonal fibrations. |
Let $r \geq 2$ be an integer, and let $a$ be an integer coprime to $r$. We show that if $c_2 \geq n + \left\lfloor \frac{r-1}{2r}a^2 + \frac{1}{2}(r^2 + 1) \right\rfloor$, then the $2n$th Betti number of the moduli space $M_{\mathbb{P}^2,H}(r,aH,c_2)$ stabilizes, where $H = c_1(\mathcal{O}_{\mathbb{P}^2}(1))$. |
This text is my thesis, defended in June 2007, in the status it was at this time. The most important results are contained in the article "Foncteur de Picard d'un champ algébrique" to appear in "Mathematische Annalen" (see the preprint <a href="https://arxiv.org/abs/0711.4545" data-arxiv-id="0711.4... |
As the sequel to [5, 7], we construct a simply connected minimal complex surface of general type with p_g = 0 and K^2 = 4 by using a rational blow-down surgery and Q-Gorenstein smoothing theory. |
In this article we give explicit formulas for the equations of a generic genus $4$ curve in terms of its theta constants. The method uses the Prym construction and the beautiful classical geometry around it. |
We prove the Conjecture of Catenese--Chen--Zhang: the inequality $K_X^3\geq \frac{4}{3}p_g(X)-\frac{10}{3}$ holds for all projective Gorenstein minimal 3-folds $X$ of general type. |
We establish Thom's jet transversality theorem for regular maps from an affine algebraic manifold to an algebraic manifold satisfying a suitable flexibility condition. It can be considered as the algebraic version of Forstnerič's jet transversality theorem for holomorphic maps from a Stein manifold to an Oka m... |
We show that the pseudoeffective cone of divisors $\overline{\text{Eff}}^1(\overline{\mathcal{M}}_{g,n})$ for $g\geq 2$ and $n\geq 2$ is not polyhedral by showing that the class of the fibre of the morphism forgetting one point forms an extremal ray of the dual nef cone of curves $\overline{\text{Nef}}_1(\overline{\mat... |
In this paper we build a link between the Teichmuller theory of hyperbolic Riemann surfaces and isomonodromic deformations of linear systems whose monodromy group is the Fuchsian group associated to the given hyperbolic Riemann surface by the Poincare' uniformization. In the case of a one-sheeted hyperboloid with ... |
We study genus zero wall-crossing for a family of moduli spaces introduced recently by Fan-Farvis-Ruan. The family has a wall and chamber structure relative to a positive rational parameter. |
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