Updated version. Based on a L. Göttsche's result we also show that generating function of the Hilbert scheme of points (0-dimensional subschemes) on a surface is an exponetial of the surface
Let $\cal R$ be either the Grothendieck semiring (semiring with multiplication) of complex algebraic varieties, or the Grothendieck ring of these varieties, or the Grothendieck ring localized by the class of the complex affine line. We introduce a construction which defines operations of taking powers of series over these (semi)rings. This means that, for a power series $A(t)=1+\sum\limits_{i=1}^\infty A_i t^i$ with the coefficients $A_i$ from $\cal R$ and for $M\in {\cal R}$, there is defined a series $(A(t))^M$ (with coefficients from $\cal R$ as well) so that all the usual properties of the exponential function hold.We also express in these terms the generating function of the Hilbert scheme of points (0-dimensional subschemes) on a surface.
We construct smooth projective varieties of general type with the smallest known volume and others with the most known vanishing plurigenera in high dimensions. The optimal volume bound is expected to decay doubly exponentially with dimension, and our examples achieve this decay rate. We also consider the analogous questions for other types of varieties. For example, in every dimension we conjecture the terminal Fano variety of minimal volume, and the canonical Calabi-Yau variety of minimal volume. In each case, our examples exhibit doubly exponential behavior.
Sometimes it is possible to extend some using Newton polyhedra computations in algebraic geometry from polynomials to exponential sums. For this purpose it is useful to consider analogues of tropical varieties in complex space. These analogues are called exponential tropical varieties (ETV). We construct the ring of ETV. Algebraic tropical varieties form the subring of the ring of ETV. In this paper we connect ETV with the complex Monge-Ampere operator action on the space of piecewise linear functions in complex vector space. We show that all ETV arise as results of such operator action. We gi…
Exponential varieties arise from exponential families in statistics. These real algebraic varieties have strong positivity and convexity properties, familiar from toric varieties and their moment maps. Among them are varieties of inverses of symmetric matrices satisfying linear constraints. This class includes Gaussian graphical models. We develop a general theory of exponential varieties. These are derived from hyperbolic polynomials and their integral representations. We compare the multidegrees and ML degrees of the gradient map for hyperbolic polynomials.
The complete first order theories of the exponential differential equations of semiabelian varieties are given. It is shown that these theories also arises from an amalgamation-with-predimension construction in the style of Hrushovski. The theory includes necessary and sufficient conditions for a system of equations to have a solution. The necessary condition generalizes Ax's differential fields version of Schanuel's conjecture to semiabelian varieties. There is a purely algebraic corollary, the "Weak CIT" for semiabelian varieties, which concerns the intersections of algebraic subgroups with …
We prove, that Hausel's formula for the number of rational points of a Nakajima quiver variety over a finite field also holds in a suitable localization of the Grothendieck ring of varieties. In order to generalize the arithmetic harmonic analysis in his proof we use Grothendieck rings with exponentials as introduced by Cluckers-Loeser and Hrushovski-Kazhdan.
We construct new six-functor formalisms capturing cohomological invariants of varieties with potentials. Starting from any six-functor formalism $C$, encoded as a coefficient system, we associate a new six-functor formalism $C_{\text{exp}}$. This requires in particular constructing the convolution product symmetric monoidal structure at the $\infty$-categorical level. We study $C_{\text{exp}}$ and how it relates to $C$. We also define motives in $C_{\text{exp}}$ attached to varieties with potential and study their properties.
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 satisfied by exp from LG to G.
We study the complexity of approximating complex zero sets of certain $n$-variate exponential sums. We show that the real part, $R$, of such a zero set can be approximated by the $(n-1)$-dimensional skeleton, $T$, of a polyhedral subdivision of $\mathbb{R}^n$. In particular, we give an explicit upper bound on the Hausdorff distance: $Δ(R,T) =O\left(t^{3.5}/δ\right)$, where $t$ and $δ$ are respectively the number of terms and the minimal spacing of the frequencies of $g$. On the side of computational complexity, we show that even the $n=2$ case of the membership problem for $R$ is undecidable i…
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. We prove explicitly that these manifolds have the structure of a pre-Frobenius manifold, being a pre-structure appearing in the process of axiomatisation of Topological Field Theory. On one hand, this allows us to develop relations to algebraic geometry, by proving explicitly that a statistical pre-Frobenius manifold forms an algebraic variety over $\mathbb{Q}$ …
The local zeta functions (also called Igusa's zeta functions) over p-adic fields are connected with the number of solutions of congruences and exponential sums mod p^{m}. These zeta functions are defined as integrals over open and compact subsets with respect to the Haar measure. In this paper, we introduce new integrals defined over submanifolds, or more generally, over non-degenerate complete intersection varieties, and study their connections with some arithmetical problems such as estimation of exponential sums mod p^{m}. In particular we extend Igusa's method for estimating exponential su…
We determine the set of connected components of closed affine Deligne-Lusztig varieties for special maximal compact subgroups of split connected reductive groups. Especially, we show that such an affine Deligne-Lusztig variety has isolated points if and only if its dimension is 0. From the dimension formula for affine Deligne-Lusztig varieties we obtain a description of the set of these varieties that are zero-dimensional.
A motivic height zeta function associated to a family of varieties parametrised by a curve is the generating series of the classes, in the Grothendieck ring of varieties, of moduli spaces of sections of this family with varying degrees. This text is devoted to the study of the motivic height zeta function associated to a family of varieties with generic fiber having the structure of an equivariant compactification of a vector group. Our main theorem describes the convergence of this motivic height zeta function with respect to a topology on the Grothendieck ring of varieties coming from the th…
Motivated by a connection between the topology of (generalized) configuration spaces and chromatic polynomials, we show that generating functions of Hodge-Deligne polynomials of quasiprojective varieties and colorings of acyclic directed graphs with the complete graph $K_n$ as the underlying undirected graph. In order to do this, we combine an interpretation of Crew-Spirkl for plethysms involving chromatic symmetric polynomials using colorings of directed acyclic graphs to plethystic exponentials often appearing Hodge-Deligne polynomials of varieties. Applying this to a recent result of Floren…
Given a semi-algebraic set S, we study compactifications of S that arise from embeddings into complete toric varieties. This makes it possible to describe the asymptotic growth of polynomial functions on S in terms of combinatorial data. We extend our earlier work to compute the ring of bounded functions in this setting and discuss applications to positive polynomials and the moment problem. Complete results are obtained in special cases, like sets defined by binomial inequalities. We also show that the wild behaviour of certain examples constructed by Krug and by Mondal-Netzer cannot occur in…
We determine the set of connected components of minuscule affine Deligne-Lusztig varieties for special maximal compact subgroups of unramified connected reductive groups. Partial results are also obtained for non-minuscule closed affine Deligne-Lusztig varieties. We consider both the function field case and its analog in mixed characteristic. In particular, we determine the set of connected components of unramified Rapoport-Zink spaces.
In this note, using methods introduced by Hacon, McKernan and Xu, we study the accumulation points of volumes of varieties of log general type. First, we show that, if the set of boundary coefficients $Λ$ is DCC, closed under limits and contains 1, then also the corresponding set of volumes is DCC and closed under limits. Then, we consider the case of $ε$-log canonical varieties, for $0 < ε< 1$. In this situation, we prove that, if $Λ$ is finite, then the corresponding set of volumes is discrete.
For a hypersurface in a projective space, we consider the set of pairs of a point and a line in the projective space such that the line intersects the hypersurface at the point with a fixed multiplicity. We prove that this set of pairs forms a smooth variety for a general hypersurface.
We examine the set of $J_b(F)$-orbits in the set of irreducible components of affine Deligne-Lusztig varieties for a hyperspecial subgroup and minuscule coweight $μ$. Our description implies in particular that its number of elements is bounded by the dimension of a suitable weight space in the Weyl module associated with $μ$ of the dual group.
We consider the set of forms of a toric variety over an arbitrary field: those varieties which become isomorphic to a toric variety after base field extension. In contrast to most previous work, we also consider arbitrary isomorphisms rather than just those that respect a torus action. We define an injective map from the set of forms of a toric variety to a non-abelian second cohomology set, which generalizes the usual Brauer class of a Severi-Brauer variety. Additionally, we define a map from the set of forms of a toric variety to the set of forms of a separable algebra along similar lines to…
We study the secant varieties of the Veronese varieties and of Veronese reembeddings of a smooth projective variety. We give some conditions, under which these secant varieties are set-theoretically cut out by determinantal equations. More precisely, they are given by minors of a catalecticant matrix. These conditions include the case when the dimension of the projective variety is at most 3 and the degree of reembedding is sufficiently high. This gives a positive answer to a set-theoretic version of a question of Eisenbud in dimension at most 3. For dimension four and higher we produce plenty…
We show how to construct a stick figure of lines in $\mathbb{P}^3$ using the Hadamard product of projective varieties. Then, applying the results of Migliore and Nagel, we use such stick figure to build a Gorenstein set of points with given $h-$vector ${\mathbf h}$. Since the Hadamard product is a coordinate-wise product, we show, at the end, how the coordinates of the points, in the Gorenstein set, can be directly determined.
A subvariety of a complex projective space has a well-known dual variety, which is the set of its tangent hyperplanes. The purpose of this paper is to generalise this notion for a subvariety of a quite general partial flag variety. A similar biduality theorem is proved, and the dual varieties of Schubert varieties are described.
We describe a class of affine toric varieties $V$ that are set-theoretically minimally defined by codim $V+1$ binomial equations over fields of any characteristic.
We present a class of homogeneous ideals which are generated by monomials and binomials of degree two and are set-theoretic complete intersections. This class includes certain reducible varieties of minimal degree and, in particular, the presentation ideals of the fiber cone algebras of monomial varieties of codimension two.
We give a necessary and sufficient condition for the nonsingular projective toric variety associated to a building set to be weak Fano in terms of the building set.
We find finite, reasonably small, generator sets of the coordinate rings of G-character varieties of finitely generated groups for all classical groups G. This result together with the method of Grobner basis gives an algorithm for describing character varieties by explicit polynomial equations. Additionally, we describe finite sets of generators of the fields of rational functions on G-character varieties for all exceptional algebraic groups G.
Given a set $Γ$ of low-degree k-dimensional varieties in $\mathbb{R}^n$, we prove that for any $D \ge 1$, there is a non-zero polynomial $P$ of degree at most $D$ so that each component of $\mathbb{R}^n \setminus Z(P)$ intersects $O(D^{k-n} |Γ|)$ varieties of $Γ$.
We study the problem of describing local components of height functions on abelian varieties over characteristic $0$ local fields as functions on spaces of torsors under various realisations of a $2$-step unipotent motivic fundamental group naturally associated to the defining line bundle. To this end, we present three main theorems giving such a description in terms of the $\mathbb Q_\ell$- and $\mathbb Q_p$-pro-unipotent étale realisations when the base field is $p$-adic, and in terms of the $\mathbb R$-pro-unipotent Betti--de Rham realisation when the base field is archimedean. In the cours…
Let $X$ and $Y$ be smooth and projective varieties over a field $k$ finitely generated over $\mathbb Q$, and let $\ov X$ and $\ov Y$ be the varieties over an algebraic closure of $k$ obtained from $X$ and $Y$, respectively, by extension of the ground field. We show that the Galois invariant subgroup of $\Br(\ov X)\oplus \Br(\ov Y)$ has finite index in the Galois invariant subgroup of $\Br(\ov X\times\ov Y)$. This implies that the cokernel of the natural map $\Br(X)\oplus\Br(Y)\to\Br(X\times Y)$ is finite when $k$ is a number field. In this case we prove that the Brauer-Manin set of the product…
To each prime ideal in a polynomial ring over a field we associate an algebraic matroid and show that it is preserved under tropicalization. This gives a necessary condition for a tropical variety to be set-theoretically realizable from a prime ideal. We also show that there are infinitely many Bergman fans that are not set-theoretically realizable as the tropicalization of any ideal.
We study the $J_b(F)$-action on the set of top-dimensional irreducible components of affine Deligne--Lusztig varieties in the affine Grassmannian. We show that the stabilizer of any such component is a parahoric subgroup of $J_b(F)$ of maximal volume, verifying a conjecture of X.~Zhu. As an application, we give a description of the set of top-dimensional irreducible components in the basic locus of Shimura varieties.
Michael Gromov has recently initiated what he calls ``symbolic algebraic geometry", in which objects are proalgebraic varieties: a proalgebraic variety is by definition the projective limit of a projective system of algebraic varieties. In this paper we construct Chern--Schwartz--MacPherson classes of proalgebraic varieties, by introducing the notion of ``proconstructible functions " and "$χ$-stable proconstructible functions" and using the Fulton-MacPherson's Bivariant Theory. As a "motivic" version of a $χ$-stable proconstructible function, $\Ga$-stable constructible functions are introduced…
We show that for every prime $p$, there is a class of Veronese varieties which are set-theoretic complete intersections if and only if the ground field has characteristic $p$.
The regular \Z^r-covers of a finite cell complex X are parameterized by the Grassmannian of r-planes in H^1(X,\Q). Moving about this variety, and recording when the Betti numbers b_1,..., b_i of the corresponding covers are finite carves out certain subsets Ω^i_r(X) of the Grassmannian. We present here a method, essentially going back to Dwyer and Fried, for computing these sets in terms of the jump loci for homology with coefficients in rank 1 local systems on X. Using the exponential tangent cones to these jump loci, we show that each Ω-invariant is contained in the complement of a union of …
We consider the problem of testing whether the points in a complex or real variety with non-zero coordinates form a multiplicative group or, more generally, a coset of a multiplicative group. For the coset case, we study the notion of shifted toric varieties which generalizes the notion of toric varieties. This requires a geometric view on the varieties rather than an algebraic view on the ideals. We present algorithms and computations on 129 models from the BioModels repository testing for group and coset structures over both the complex numbers and the real numbers. Our methods over the comp…
From a finite set in a lattice, we can define a toric variety embedded in a projective space. In this paper, we give a combinatorial description of the dual defect of the toric variety using the structure of the finite set as a Cayley sum with suitable conditions. We also interpret the description geometrically.
In the theory of algebraic group actions on affine varieties, the concept of a Kempf-Ness set is used to replace the categorical quotient by the quotient with respect to a maximal compact subgroup. By making use of the recent achievements of "toric topology" we show that an appropriate notion of a Kempf-Ness set exists for a class of algebraic torus actions on quasiaffine varieties (coordinate subspace arrangement complements) arising in the Batyrev-Cox "geometric invariant theory" approach to toric varieties. We proceed by studying the cohomology of these "toric" Kempf-Ness sets. In the case …
Normal toric varieties over a field or a discrete valuation ring are classified by rational polyhedral fans. We generalize this classification to normal toric varieties over an arbitrary valuation ring of rank one. The proof is based on a generalization of Sumihiro's theorem to this non-noetherian setting. These toric varieties play an important role for tropicalizations.
We study the set of connected components of certain unions of affine Deligne-Lusztig varieties arising from the study of Shimura varieties. We determine the set of connected components for basic $\s$-conjugacy classes. As an application, we verify the Axioms in \cite{HR} for certain PEL type Shimura varieties. We also show that for any nonbasic $\s$-conjugacy class in a residually split group, the set of connected components is "controlled" by the set of straight elements associated to the $\s$-conjugacy class, together with the obstruction from the corresponding Levi subgroup. Combined with \…
For a given irreducible projective variety $X$, the closure of the set of all hyperplanes containing tangents to $X$ is the projectively dual variety $X^{\vee}$. We study the singular locus of projectively dual varieties of certain Segre-Plücker embeddings in series of papers. In this work we give a classification of the irreducible components of the singular locus of the dual Grassmannian. Basically, it admits two components: cusp type and node type which are degeneracies of a certain Hessian matrix, and the closure of the set of tangent planes having more than one critical point, respectivel…
We introduce and study the notion of affine varieties associated to ordered bases and establish Galois connection between the power set of $A^n_K$ and the power set of $K[x_1, . . ., x_n]$, and then induce a Galois correspondence. We generalize the idea by defining affine varieties associated to linear operators. We produce Hilbert's Nullstellensatz version for such varieties and show that there is a 1-1 correspondence between this kind of varieties in $A^n_K$ and the "usual" affine varieties in $A^n_K$. We prove that the \usual" affine varieties forms a skeleton for the category of all affine…
We generalize the Guth--Katz joints theorem from lines to varieties. A special case says that $N$ planes (2-flats) in 6 dimensions (over any field) have $O(N^{3/2})$ joints, where a joint is a point contained in a triple of these planes not all lying in some hyperplane. More generally, we prove the same bound when the set of $N$ planes is replaced by a set of 2-dimensional algebraic varieties of total degree $N$, and a joint is a point that is regular for three varieties whose tangent planes at that point are not all contained in some hyperplane. Our most general result gives upper bounds, tig…
We introduce certain quiver analogue of the determinantal variety. We study the Kempf-Lascoux-Weyman's complex associated to a line bundle on the variety. In the case of generalized Kronecker quivers, we give a sufficient condition on when the complex resolves a maximal Cohen-Macaulay module supported on the quiver determinantal variety. This allows us to find the set-theoretical defining equations of these varieties. When the variety has codimension one, the only irreducible polynomial function is a relative tensor invariant. As a by-product, we find some vanishing condition for the Kronecker…
