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We compare two known definitions for a relative family of effective zero cycles, based on traces and norms of functions, respectively. In characteristic zero we show that both definitions agree. |
In this paper we investigate the space of $\mathbb{R}$-places of an algebraic function field of one variable. We deal with the problem of determining when two orderings of such a field correspond to a single $\mathbb{R}$-place. |
We study a period map for triple coverings of $\mathbf P^2$branching along special configurations of $6$ lines. Though the moduli space of special configurations isa two dimensional variety,the minimal models of the coverings form a oneparameter family of K3 <a href="http://surfaces.We" rel="external noopener nofollow... |
Let $A$, $B$ be multidimensional matrices of boundary format respectively $\prod_{i=0}^p(k_i+1)$, $\prod_{j=0}^q(l_j+1)$. Assume that $k_p=l_0$ so that the convolution $A\ast B$ is defined. |
We prove that the pull-back of a quasi-log scheme by a smooth quasi-projective morphism has a natural quasi-log structure. We treat an application to log Fano pairs. |
The purpose of this note is to point out an elementary but somewhat surprising connection between the work of Buser and Sarnak on lengths of periods of abelian varieties and the Seshadri constants measuring the local positivity of theta divisors. The link is established via symplectic blowing up, in the spirit of McDu... |
We introduce the notion of a quasi DG category, generalizing that of a DG category. To a quasi DG category satisfying certain additional conditions, we associate another quasi DG category, the quasi DG category of $C$-diagrams. |
This paper presents a gentle introduction to cohomology vanishing theorems, largely based on the paper work of Hongshan Li. It offers an insightful exploration of unitary local systems on complex manifolds, particularly focusing on their characteristics near normal crossing divisors. |
Consider a linear realization of a matroid over a field. One associates with it a configuration polynomial and a symmetric bilinear form with linear homogeneous coefficients. |
We initiate a study of path spaces in the nascent context of "motivic dga's", under development in doctoral work by Gabriella Guzman. This enables us to reconstruct the unipotent fundamental group of a pointed scheme from the associated augmented motivic dga, and provides us with a factorization of Kim'... |
Let $C$ be a projective smooth connected curve over an algebraically closed field of characteristic zero, let $F$ be its field of functions, let $C_0$ be a dense open subset of $C$. Let $X$ be a projective flat morphism to $C$ whose generic fiber $X_F$ is a smooth equivariant compactification of $G$ such that $D=X_F\s... |
Let X be a smooth complex manifold. Let Sol denote the solution functor for D-modules on X. Traditionally, the fully-faithfulness of Riemann-Hilbert correspondance is proved by showing that if M_1 and M_2 are regular holonomic D_X modules, then the canonical morphism of complexes of sheaves RH_{M_1,M_2} : RHom(M_1,M_2... |
In this paper we give a birational model for the theta divisor of the intermediate Jacobian of a generic cubic threefold $X$. We use the standard realization of $X$ as a conic bundle and a $4-$dimensional family of plane quartics which are totally tangent to the discriminant quintic curve of such a conic bundle struct... |
We prove that the $\infty$-category of motivic spectra satisfies Milnor excision: if $A\to B$ is a morphism of commutative rings sending an ideal $I\subset A$ isomorphically onto an ideal of $B$, then a motivic spectrum over $A$ is equivalent to a pair of motivic spectra over $B$ and $A/I$ that are identified over $B/I... |
In this paper, we study the capped vertex functions associated to certain zero-dimensional type-$A$ Nakajima quiver varieties. The insertion of descendants into the vertex functions can be expressed by the Macdonald operators, which leads to explicit combinatorial formulas for the capped vertex functions. |
We construct the semi-infinite tensor structure on the semiderived category of quasi-coherent torsion sheaves on an ind-scheme endowed with a flat affine morphism into an ind-Noetherian ind-scheme with a dualizing complex. The semitensor product is "a mixture of" the cotensor product along the base and the der... |
Let $S\to C$ be a smooth quasi-projective surface properly fibered onto a smooth curve. We prove that the multiplicativity of the perverse filtration on $H^*(S^{[n]},\mathbb{Q})$ associated with the natural map $S^{[n]}\to C^{(n)}$ implies that $S\to C$ is an elliptic fibration. |
We study the Betti tables of reducible algebraic curves with a focus on connected line arrangements and provide a general formula for computing the quadratic strand of the Betti table for line arrangements that satisfy certain hypotheses. We also give explicit formulas for the entries of the Betti tables for all curve... |
We show that the equivariant Gromov-Witten invariants of a projective homogeneous space G/P exhibit Graham-positivity: when expressed as polynomials in the positive roots, they have nonnegative coefficients. |
In this work we investigate partition models, the subset of log-linear models for which one can perform the iterative proportional scaling (IPS) algorithm to numerically compute the maximum likelihood estimate (MLE). Partition models include families of models such as hierarchical models and balanced, stratified stage... |
We give a combinatorial algorithm for equivariant embedded resolution of singularities of a toric variety defined over a perfect field. The algorithm is realized by a finite succession of blowings-up with smooth invariant centres that satisfy the normal flatness criterion of Hironaka. |
In this paper we study twisted complexes on a ringed space and prove that it gives a new dg-enhancement of the derived category of perfect complexes on that space. A twisted complex is a collection of locally defined sheaves together with the homotopic gluing data. |
We study from a geographical point of view fibrations of threefolds over smooth curves, such that the general fibre is of general type. We prove the non-negativity of certain relative invariants under general hypotheses and give lower bounds for the self-interssection of the relative canonical divisor of the fibration... |
Let $S$ be a smooth algebraic surface in $\mathbb{P}^3(\mathbb{C})$. A curve $C$ in $S$ has a cohomology class $\eta_C \in H^1 \hspace{-3pt}\left( \Omega^1_S \right)$. |
We explicitly determine the elliptic K3 surfaces with a maximal singular fibre. If the characteristic of the ground field is different from 2, for each of the two possible maximal fibre types, $I_{19}$ and $I^*_{14}$, the surface is unique. |
Motivated by gauge theory on manifolds with exceptional holonomy, we construct examples of stable bundles on K3 surfaces that are invariant under two involutions: one is holomorphic; and the other is anti-holomorphic. These bundles are obtained via the monad construction, and stability is examined using the Generalise... |
We show that the space of theta functions on tropical tori is identified with a convex polyhedron. We also show a Riemann-Roch inequality for tropical abelian surfaces by calculating the self-intersection numbers of divisors. |
The F-conjecture gives a conjectural description of the ample cone of the Deligne-Mumford moduli space $\overline{M}_{g,n}$. We prove that the $S_n$-symmetric and the non-symmetric F-conjectures are equivalent. |
Chen-Gounelas-Liedtke recently introduced a powerful regeneration technique, a process opposite to specialization, to prove existence results for rational curves on projective $K3$ surfaces. We show that, for projective irreducible holomorphic symplectic manifolds, an analogous regeneration principle holds and provide... |
Given a compact complex algebraic variety with an effective action of a finite group $G$, and a class $\alpha \in H^2(G,U(1))$, we introduce an orbifold elliptic genus with discrete torsion $\alpha$, denoted $Ell^{\alpha}_{orb}(X,G, q, y)$. We give an interpretation of this genus in terms of the chiral de Rham complex... |
It is a long-established and heavily-used fact that the integral cohomology ring of the Deligne-Mumford moduli space of (complex) rational curves is the polynomial ring on the boundary divisors modulo the ideal generated by the obvious geometric relations between them. We show that the rational cohomology ring of the ... |
We factor the virtual Poincare polynomial of every homogeneous space G/H, where G is a complex connected linear algebraic group and H is an algebraic subgroup, as $t^{2u} (t^2-1)^r Q_{G/H}(t^2)$ for a polynomial $Q_{G/H}$ with non-negative integer coefficients. Moreover, we show that $Q_{G/H}(t^2)$ divides the virtual... |
We prove that the enumerative geometry of lines on smooth cubic surfaces is governed by the arithmetic of the base field. In 1949, Segre proved that the number of lines on a smooth cubic surface over any field is 0, 1, 2, 3, 5, 7, 9, 15, or 27. |
We prove that the geometric Bogomolov conjecture holds for nowhere degenerate abelian varieties of dimension $5$ with trivial trace. By this result together with our previous work, we see that the conjecture holds for an abelian variety such that the difference between the dimension of its maximal nowhere degenerate a... |
We develop an approach to noncommutative algebraic geometry ``in the perturbative regime" around ordinary commutative geometry. Let R be a noncommutative algebra and A=R/[R,R] its commutativization. |
A method of constructing algebraic-geometric codes with many automorphisms arising from Galois points for algebraic curves is presented. |
We prove an analogue in Arakelov geometry of the Grothendieck-Riemann-Roch theorem. |
We prove that the action of the special automorphism group on affine cones over del Pezzo surfaces of degree 4 and 5 is infinitely transitive. |
We study the monodromies and the limit mixed Hodge structures of families of complete intersection varieties over a punctured disk in the complex plane. For this purpose, we express their motivic nearby fibers in terms of the geometric data of some Newton polyhedra. |
After results by the author (1980, 1981), and by Vinberg (1981), finiteness of the number of maximal arithmetic reflection groups in Lobachevsky spaces was not known in dimensions $2\le n\le 9$ only. <br>Recently (2005), the finiteness was proved in dimension 2 by Long, Maclachlan and Reid, and in dimension 3 by Agol.... |
We present an algorithm for computing equations of canonically embedded Riemann surfaces with automorphisms. A variant of this algorithm with many heuristic improvements is used to produce equations of Riemann surfaces $X$ with large automorphism groups (that is, $|\mathrm{Aut}(X)| > 4(g_X-1)$) for genus $4 \leq g_... |
We study intersection cohomology of moduli spaces of semistable vector bundles on a complex algebraic surface. Our main result relates intersection Poincare polynomials of the moduli spaces to Donaldson-Thomas invariants of the surface. |
A bracket is a function that assigns a number to each monomial in variables \tau_0, \tau_1, ... We show that any bracket satisfying the string and the dilaton relations gives rise to a power series lying in the algebra A generated by the series \sum n^{n-1} q^n/n! and \sum n^n q^n /n! |
Given a Kahler group, we study the set of homomorphisms from this group to the mapping class group which can be realized as the monodromy of a holomorphic family of curves. |
In this paper, using what we call a micro reciprocity law, we complete Weil's program for non-abelian class field theory of Riemann surfaces. |
A well-known result of Murray Marshall states that every $f \in \mathbb{R} [X,Y]$ non-negative on the strip $[0,1] \times \mathbb{R}$ can be written as $f= \sigma_0 + \sigma_1 X(1-X)$ with $\sigma_0, \sigma_1$ sums of squares in $\mathbb{R} [X,Y]$. In this work, we present a few results concerning this representation ... |
The point is to compare the mathematical meaning of the ``number of rational curves on a Calabi-Yau threefold'' to the meaning ascribed to the same notion by string theorists. |
Two different constructions of an invariant of an odd dimensional hyperbolic manifold in the K-group $K_{2n-1}(\bar \Bbb Q)\otimes \Bbb Q$ are given. The volume of the manifold is equal to the value of the Borel regulator on that element. |
Let $f:X\to Y$ be a proper, dominant morphism of smooth varieties over a number field $k$. When is it true that for almost all places $v$ of $k$, the fibre $X_P$ over any point $P\in Y(k_v)$ contains a zero-cycle of degree $1$? |
We propose a linear version of the weighted bounded negativity conjecture. It considers a smooth projective surface $X$ over an algebraically closed field of characteristic zero and predicts the existence of a common lower bound on $C^2/(D\cdot C)$ for all reduced and irreducible curves $C$ and all big and nef divisor... |
An old conjecture of Voisin describes how zero-cycles on a variety $X$ should behave when pulled-back to the self-product $X^m$ for $m$ larger than the geometric genus of $X$. Using complete intersections of quadrics, we give examples of varieties in any dimension and with arbitrarily high geometric genus that verify ... |
The results in this paper imply that for every number field F and positive integer r, there exists an F-isogeny class of abelian varieties such that r divides the degree of every F-polarization on every abelian variety in the isogeny class. |
We employ the formalism of vanishing cycles and perverse sheaves to introduce and study the vanishing cohomology of complex projective hypersurfaces. As a consequence, we give upper bounds for the Betti numbers of projective hypersurfaces, generalizing those obtained by different methods by Dimca in the isolated singu... |
The Orlov spectrum is a new invariant of a triangulated category. It was introduced by D. Orlov building on work of A. Bondal-M. |
For a generic hypersurface $\mathbb{X}^{n-1} \subset \mathbb{P}^n(\mathbb{C})$ of degree \[ d \,\geqslant\, n^{2n} \] (1) $\mathbb{P}^n \big\backslash \mathbb{X}^{n-1}$ is Kobayashi-hyperbolically imbedded in $\mathbb{P}^n$; <br>(2) $\mathbb{X}^{n-1}$ is Kobayashi($\Leftrightarrow$ Brody)-hyperbolic. <br>(1) improves ... |
This paper is an introduction to the jet schemes and the arc space of an algebraic variety. We also introduce the Nash problem on arc families. |
We show that the family of 21-dimensional intermediate jacobians of cubic fivefolds containing a given cubic fourfold X is generically an algebraic integrable system. In the proof we apply an integrability criterion, introduced and used by Donagi and Markman to find a similar integrable system over the family of cubic... |
In this paper, we express Euler's formula and De Moivre's formula for Clifford algebra Cl2 and find nth roots of an element in Clifford algebra Cl2. |
We consider the algebro-geometric consequences of integration by parts. |
We derive some combinatorial consequences from the positivity of Donaldson-Thomas invariants for symmetric quivers conjectured by Kontsevich and Soibelman and proved recently by Efimov. These results are used to prove the Kac conjecture for quivers having at least one loop at every vertex. |
We prove a logarithmic base change theorem for pushforwards of pluri-canonical bundles and use it to deduce that positivity properties of log canonical divisors descend via smooth projective morphisms. As an application, for a surjective morphism $f:X\to Y$ with $\kappa(X)\ge 0$ and $-K_Y$ big, we prove $Y\setminus \D... |
We introduce the notion of a regular integrable connection on a smooth log scheme over $\mathbf{C}$ and construct an equivalence between the category of such connections and the category of integrable connections on its analytification, compatible with de Rham cohomology. This extends the work of Deligne (when the log... |
We define a proper moduli stack for degree $p$ covers $f:Y \to \cX$ where $\cX$ is a twisted stable curve in the sense of [5] and [4], and $Y$ is a stable curve which via $f$ is a torsor over $\cX$ under a finite flat group scheme $\cG \to \cX$. |
Let $L$ and $L'$ be two invertible sheaves over a projective variety $X$. We suppose that $L$ and $L'$ are generated by their global section spaces $\Gamma(L)$ and $\Gamma(L')$. |
We show that there is a pair of homogeneous polynomials such that the sets of roots of their Bernstein-Sato polynomials which are strictly supported at the origin are different although the sets of roots which are not strictly supported at the origin are the same and moreover their graded Milnor algebras have the same ... |
Starting from the classical division polynomials we construct homogeneous polynomials $\alpha_n$, $\beta_n$, $\gamma_n$ such that for $P = (x:y:z)$ on an elliptic curve in Weierstrass form over an arbitrary ring we have $nP = \bigl(\alpha_n(P):\beta_n(P):\gamma_n(P)\bigr)$. To show that $\alpha_n,\beta_n,\gamma_n$ ind... |
We give a light introduction to some recent developments in Mori theory, and to our recent direct proof of the finite generation of the canonical ring. |
Main topic of the paper is the determination, for a compact complex manifold $M$, of the class of manifolds $X$ which are deformation equivalent to it. If $M$ is a complex torus, then also $X$ is so. |
We introduce a new obstruction to lifting smooth proper varieties in characteristic $p>0$ to characteristic $0$. It is based on Grothendieck's specialization homomorphism and the resulting discrete finiteness properties of étale fundamental groups. |
Let $C$ be a curve over a non-archimedean local field of characteristic zero. We formulate algebro-geometric statements that imply boundedness of functions on the moduli space of stable bundles of rank $2$ and fixed odd degree determinant over $C$, coming from the Schwartz space of $\kappa$-densities on the correspond... |
Recent work of Ein-Lazarsfeld-Smith and Hochster-Huneke raised the problem of determining which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci-Harbourne defined a quantity called the resurgence to address this problem for homogeneous ideals in polynomial rings, with a focus on ... |
In this paper, we give a simple description of the deformations of a map between two smooth curves with partially prescribed branching, in the cases that both curves are fixed, and that the source is allowed to vary. Both descriptions work equally well in the tame or wild case. |
For a certain class of vector bundles E on abelian varieties A over local fields containing all line bundles algebraically equivalent to zero we define a canonical representation of the Tate module of A on the fibre of E in the zero section. This extends an old construction of Tate for line bundles to vector bundles o... |
We construct several examples of higher-dimensional Calabi-Yau manifolds and prove their modularity. |
We complete Mori's program with symmetric divisors for the moduli space of stable seven pointed rational curves. We describe all birational models in terms of explicit blow-ups and blow-downs. |
In an article from 1865, Arthur Cayley claims that given a plane algebraic curve there exists an associated 2-Hessian curve that intersects it in its sextactic points. In this paper we fix an error in Cayley's calculations and provide the correct defining polynomial for the 2-Hessian. |
We prove a formula expressing the Log Gromov-Witten Invariants of a product of log smooth varieties $V \times W$ in terms of the invariants of $V$ and $W$. This extends results of F. Qu and Y.P. Lee, who introduced this formula analogously to K. Behrend. |
We consider a rather special class of translation surfaces (called M-Origamis in this work) that are obtained from dessins by a construction introduced by Martin Möller. We give a new proof with a more combinatorial flavour of Möller's theorem that $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts faithfully o... |
We define, in a purely algebraic way, 1-motives $Alb^{+}(X)$, $Alb^{-}(X)$, $Pic^{+}(X)$ and $Pic^{-}(X)$ associated with any algebraic scheme $X$ over an algebraically closed field of characteristic zero. For $X$ over $\C$ of dimension $n$ the Hodge realizations are, respectively, $H^{2n-1}(X)(n)$, $H_{1}(X)$, $H^{1}... |
Given a brane tiling, that is a bipartite graph on a torus, we can associate with it a quiver potential and a quiver potential algebra. Under certain consistency conditions on a brane tiling, we prove a formula for the Donaldson-Thomas type invariants of the moduli space of framed cyclic modules over the corresponding... |
New relations between algebraic geometry, information theory and Topological Field Theory are developed. One considers models of databases subject to noise i.e. probability distributions on finite sets, related to exponential families. |
Let $X$ be a normal projective variety. A surjective endomorphism $f:X\to X$ is int-amplified if $f^\ast L - L =H$ for some ample Cartier divisors $L$ and $H$. |
We study a space of genus $g$ stable, $n$-marked tropical curves with total edge length $1$. Its rational homology is identified both with top-weight cohomology of the complex moduli space $M_{g,n}$ and with the homology of a marked version of Kontsevich's graph complex, up to a shift in degrees. |
Let $k$ be a perfect field and let $X\subset {\mathbb P}^N$ be a hypersurface of degree $d$ defined over $k$ and containing a linear subspace $L$ defined over an algebraic closure $\overline{k}$ with $\mathrm{codim}_{{\mathbb P}^N}L=r$. We show that $X$ contains a linear subspace $L_0$ defined over $k$ with $\mathrm{c... |
We prove that every orbit of the adjoint representation of any connected reductive algebraic group $G$ is a rational algebraic variety. For complex simply connected semisimple $G$, this implies rationality of affine Hamiltonian $G$-varieties (which we classify). |
We study intersection theory and Chern classes of reflexive sheaves on normal varieties. In particular, we define generalization of Mumford's intersection theory on normal surfaces to higher dimensions. |
Let $K$ be a complete discretely valued field with the residue field $\kappa$. Assume that cohomological dimension of $\kappa$ is less than or equal to $1$ (for example, $\kappa$ is an algebraically closed field or a finite field). |
We study finite groups of automorphisms of the $p$-adic open disk. In particular, we generalize results of Green, Matignon and Henrio from cyclic groups of order $p$ to arbitrary finite groups. |
This paper extends previous results of the authors, concerning the behaviour of the equimultiple locus of algebroid surfaces under blowing--up, to arbitrary characteristic. |
Real algebraic geometry is the study of semi-algebraic sets, subsets of $\R^k$ defined by Boolean combinations of polynomial equalities and inequalities. The focus of this thesis is to study quantitative results in real algebraic geometry, primarily upper bounds on the topological complexity of semi-algebraic sets as ... |
We prove the geometric Satake equivalence for étale metaplectic covers of reductive group schemes and extend the Langlands parametrization of V. Lafforgue to genuine cusp forms defined on their associated covering groups. |
We show that for a smooth projective variety $X$ over a field $k$ and a reduced effective Cartier divisor $D \subset X$, the Chow group of 0-cycles with modulus $\mathrm{CH}_0(X|D)$ coincides with the Suslin homology $H^S_0(X \setminus D)$ under some necessary conditions on $k$ and $D$. We derive several consequences,... |
In this work we study the monodromy group of covers $\varphi \circ \psi$ of curves \linebreak $\mathcal{Y}\xrightarrow {\quad {\psi}} \mathcal{X} \xrightarrow {\quad \varphi} \mathbb{P}^{1}$, where $\psi$ is a $q$-fold cyclic étale cover and $\varphi$ is a totally ramified $p$-fold cover, with $p$ and $q$ different pri... |
Let M_g be the moduli space of stable curves of genus g >= 2. Let D_i be the irreducible component of the boundary of M_g such that general points of D_i correspond to stable curves with one node of type i. |
We study deformations of irreducible Hermitian symmetric spaces $S$ of the compact type, known to be locally rigid, as projective-algberaic manifolds and prove that no jump of complex structures can occur. For each $S$ of rank $\ge 2$ there is an associated reductive linear group $G$ such that $S$ admits a holomorphic... |
We describe in terms of the j-invariant all elliptic surfaces pi: X -> C with a section, such that h^{1,1}(X)=rank NS(X) and the Mordell-Weil group of pi is finite. <br>We use this to give a complete solution to infinitesimal Torelli for elliptic surfaces with a section over P^1. |
In this paper we want to show, that the finite impulse response quadratic mirror filters (QMF) associated to a tower of grids $\Gamma\subset H=\bf Z^n$ can be identified with a right coset of the subgroup Fix$(T_{\Gamma^{\perp}}$,Map$ ({\bf T}^n\to U(N)$: poly) of the group of polynomial loops Map$({\bf T}^n\to U(N)$: ... |
A torsion-free sheaf $E$ on a projective variety $X$ is called quasi-trivial if $E^{\vee\vee}=\mathcal{O}_{X}^{\oplus r}$. While such sheaves are always $\mu$-semistable, they may not be semistable. |
Let $S$ be the (minimal) Enriques surface obtained from the symmetric quartic surface $(\sum_{i<j}x_ix_j)^2=kx_1x_2x_3x_4$ in $\mathbb{P}^3$ with $k\neq 0,4,36$, by taking quotient of the Cremona action $(x_i) \mapsto (1/x_i)$. The automorphism group of $S$ is a semi-direct product of a free product $\mathcal{F}$ o... |
We give a constructive proof of the Hodge conjecture for complex $K3$ surfaces that does not rely on Torelli-type results. Starting with an arbitrary rational $(1,1)$-class $\alpha\in H^{1,1}(X,\mathbb{Q})$, we algorithmically build a one-parameter family of quartic $K3$'s acquiring at most ten $A_1$-nodes. |
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