Irreducible Variety Research Articles

Mixed Volume and an Extension of Intersection Theory of Divisors

Let K(X) be the collection of all non-zero finite dimensional subspaces of rational functions on an n-dimensional irreducible variety X. For any n-tuple L_1,..., L_n in K(X), we define an intersection index [L_1,..., L_n] as the number of solutions in X of a system of equations f_1 = ... = f_n = 0 where each f_i is a generic function from the space L_i. In counting the solutions, we neglect the solutions x at which all the functions in some space L_i vanish as well as the solutions at which at least one function from some subspace L_i has a pole. The collection K(X) is a commutative semigroup with respect to a natural multiplication. The intersection index [L_1,..., L_n] can be extended to the Grothendieck group of K(X). This gives an extension of the intersection theory of divisors. The extended theory is applicable even to non-complete varieties. We show that this intersection index enjoys all the main properties of the mixed volume of convex bodies. Our paper is inspired by the Bernstein-Kushnirenko theorem from the Newton polytope theory.

On the space of morphisms into generic real algebraic varieties

We introduce a notion of generic real algebraic variety and we study the space of morphisms into these varieties. Let Z be a real algebraic variety. We say that Z is generic if there exist a finite family {Di }n i=1 of irreducible real algebraic curves with genus ≥ 2 and a biregular embedding of Z into the product variety ∏n i=1 Di . A bijective map φ : Z → Z from a real algebraic variety Z to Z is called weak change of the algebraic structure of Z if it is regular and its inverse is a Nash map. Generic real algebraic varieties are “generic” in the sense specified by the following result: For each real algebraic variety Z and for integer k, there exists an algebraic family {φt : Zt → Z}t∈Rk of weak changes of the algebraic structure of Z such that Z0 = Z , φ0 is the identity map on Z and, for each t ∈ Rk {0}, Zt is generic. Let X and Y be nonsingular irreducible real algebraic varieties. Regard the set R(X, Y ) of regular maps from X to Y as a subspace of the corresponding set N (X, Y ) of Nash maps, equipped with the C∞ compact-open topology. We prove that, if Y is generic, then R(X, Y ) is closed and nowhere dense in N (X, Y ), and has a semi-algebraic structure. Moreover, the set of dominating regular maps from X to Y is finite. A version of the preceding results in which X and Y can be singular is given also. Mathematics Subject Classification (2000): 14P05 (primary); 14P20 (secondary).

On the singular locus of Grassmann secant varieties

Let $X\subset \mathbb{P}^N$ be an irreducible non-degenerate variety. If the $(h,k)$-Grass\-mann secant variety $G_{h,k}(X)$ of $X$ is not the whole Grassmannian $\mathbb{G}(h,N)$, we have that the singular locus of $G_{h,k}(X)$ contains $G_{h,k-1}(X)$. Moreover, if $X$ is a smooth curve without $(2k+2)$-secant $2k$-space divisors, we obtain the equality $\text{Sing}(G_{h,k}(X))=G_{h,k-1}(X)$.

On principal bundles over a projective variety defined over a finite field

AbstractLet M be a geometrically irreducible smooth projective variety, defined over a finite field k, such that M admits a k-rational point x0. Let (M,x0/ denote the corresponding fundamental group-scheme introduced by Nori. Let EG be a principal G-bundle over M, where G is a reduced reductive linear algebraic group defined over the field k. Fix a polarization ξ on M. We prove that the following three statements are equivalent:1. The principal G-bundle EG over M is given by a homomorphism (M,x0)→G.2. There are integers b > a ≥ 1, such that the principal G-bundle (FbM)* EG is isomorphic to (FaM) * EG where FM is the absolute Frobenius morphism of M.3. The principal G-bundle EG is strongly semistable, the degree(c2(ad(EG))c1 (ξ)d−2 = 0, where d = dimM, and the degree(c1(EG(χ))c1(ξ)d−1) = 0 for every character χ of G, where EG(χ) is the line bundle over M associated to EG for χ.In [16], the equivalence between the first statement and the third statement was proved under the extra assumption that dimM = 1 and G is semisimple.

The 2-factoriality of the O’Grady moduli spaces

The aim of this work is to show that the moduli space M10 introduced by O’Grady is a 2-factorial variety. Namely, M10 is the moduli space of semistable sheaves with Mukai vector v: = (2, 0, −2) in \({H^{ev}(X,\mathbb{Z})}\) on a projective K3 surface X. As a corollary to our construction, we show that the Donaldson morphism gives a Hodge isometry between \({v^{\perp}}\) (sublattice of the Mukai lattice of X) and its image in \({H^{2} (\widetilde{M}_{10}, \mathbb{Z})}\), lattice with respect to the Beauville form of the 10-dimensional irreducible symplectic manifold \({\widetilde{M}_{10}}\), obtained as symplectic resolution of M10. Similar results are shown for the moduli space M6 introduced by O’Grady to produce its 6-dimensional example of irreducible symplectic variety.

Algebraic Hamiltonian actions

In this paper we deal with a Hamiltonian action of a reductive algebraic group G on an irreducible normal affine Poisson variety X. We study the quotient morphism $${\mu_{G,X}//G : X//G \rightarrow \mathfrak{g} //G}$$ of the moment map $${\mu_{G,X} : X\rightarrow \mathfrak{g}}$$ . We prove that for a wide class of Hamiltonian actions (including, for example, actions on generically symplectic varieties) all fibers of the morphism μ G,X //G have the same dimension. We also study the “Stein factorization” of μ G,X //G. Namely, let C G,X denote the spectrum of the integral closure of $${\mu_{G,X}^{*}(\mathbb{K}[\mathfrak{g}]^G)}$$ in $${\mathbb{K}(X)^G}$$ . We investigate the structure of the $${\mathfrak{g} //G}$$ -scheme C G,X . Our results partially generalize those obtained by F. Knop for the actions on cotangent bundles and symplectic vector spaces.

Flops on holomorphic symplectic fourfolds and determinantal cubic hypersurfaces

AbstractWe study the birational geometry of irreducible holomorphic symplectic varieties arising as varieties of lines of general cubic fourfolds containing a cubic scroll. We compute the ample and moving cones, and exhibit a birational automorphism of infinite order explaining the chamber decomposition of the moving cone.

THE DIRICHLET PROBLEM FOR MAXIMAL PLURISUBHARMONIC FUNCTIONS ON ANALYTIC VARIETIES IN ℂn

Let Ω be a B-regular domain in ℂn and let V be a locally irreducible analytic variety in Ω. Given a continuous function [Formula: see text], we prove that there is a unique maximal plurisubharmonic function u on V with boundary values given by ϕ and furthermore that u is continuous on [Formula: see text].

Calogero–Moser spaces over algebraic curves

In these notes, we give a survey of the main results of [5] and [7]. Our aim is to generalize the geometric classification of (left) ideals of the first Weyl algebra \({A_{1}(\mathbb{C})}\) (see [8, 9]) to the ring \({\mathcal{D}}(X)\) of differential operators on an arbitrary complex smooth affine curve X. We approach this problem in two steps: first, we classify the ideals of \({\mathcal{D}}(X)\) up to stable isomorphism, in terms of the Picard group of X; then, we refine this classification by describing each stable isomorphism class as a disjoint union of (quotients of) generalized Calogero–Moser spaces \({\mathcal{C}}_{n}(X,{\mathcal{I}})\). The latter are defined as representation varieties of deformed preprojective algebras over a certain noncommutative extension of the ring of regular functions on X. As in the classical case, \({\mathcal{C}}_{n}(X,{\mathcal{I}})\) turn out to be smooth irreducible varieties of dimension 2n.

G-biliaison of ladder pfaffian varieties

The ideals generated by pfaffians of mixed size contained in a subladder of a skew-symmetric matrix of indeterminates define arithmetically Cohen–Macaulay, projectively normal, reduced and irreducible projective varieties. We show that these varieties belong to the G-biliaison class of a complete intersection. In particular, they are glicci.

On pairs of commuting nilpotent matrices

Let B be a nilpotent matrix and suppose that its Jordan canonical form is determined by a partition λ. Then it is known that its nilpotent commutator \( \mathcal{N}_B\) is an irreducible variety and that there is a unique partition μ such that the intersection of the orbit of nilpotent matrices corresponding to μ with \( \mathcal{N}_{B}\) is dense in \( \mathcal{N}_{B}\). We prove that map \( \mathcal{D} \) given by \( \mathcal{D}{\left( \lambda \right)} = \mu \) is an idempotent map. This answers a question of Basili and Iarrobino [9] and gives a partial answer to a question of Panyushev [18]. In the proof, we use the fact that for a generic matrix \( A \in \mathcal{N}_{B}\) the algebra generated by A and B is a Gorenstein algebra. Thus, a generic pair of commuting nilpotent matrices generates a Gorenstein algebra. We also describe \( \mathcal{D}{\left( \lambda \right)} \) in terms of λ if \( \mathcal{D}{\left( \lambda \right)} \) has at most two parts.

Cohomological characterisation of Steiner bundles

A vector bundle E on a smooth irreducible algebraic variety X is called a Steiner bundle of type ( F 0 , F 1 ) if it is defined by an exact sequence of the form where s, t ≥ 1 and ( F 0 , F 1 ) is a strongly exceptional pair of vector bundles on X such that is generated by global sections. Let X be a smooth irreducible projective variety of dimension n with an n -block collection , of locally free sheaves on X which generate D b (𝒪 X – mod ). We give a cohomological characterisation of Steiner bundles of type on X , where 0 ≤ a < b ≤ n and 1 ≤ i 0 ≤ α a , 1 ≤ j 0 ≤ α b .

Monodromy of a family of hypersurfaces

Soit Y une variété projective complexe lisse irréductible de dimension m+1, plongée dans un espace projectif. Soit Z un sous-schéma fermé de Y, et soit δ un entier positif tel que ℐ Z,Y (δ) soit engendré par ses sections globales. Fixons un entier d≥δ+1, et supposons que le diviseur général X∈|H 0 (Y,ℐ Z,Y (d))| soit lisse. Désignons par H m (X;ℚ) ⊥Z van le quotient de H m (X;ℚ) par la cohomologie de Y et par les classes des composantes irréductibles de Z de dimension m. Dans cet article, nous prouvons que la représentation de monodromie sur H m (X;ℚ) ⊥Z van pour la famille des diviseurs lisses X∈|H 0 (Y,ℐ Z,Y (d))| est irréductible.

On the fundamental group-scheme

We show that the fundamental group-scheme of a separably rationally connected variety defined over an algebraically closed field is trivial. Let X be a geometrically irreducible smooth projective variety defined over a finite field k admitting a k-rational point. Let { E n , σ n } n ⩾ 0 be a flat principal G-bundle over X, where G is a reductive linear algebraic group defined over k. We show that there is a positive integer a such that the principal G-bundle ( F X a ) ∗ E 0 → X is isomorphic to E 0 , where F X is the absolute Frobenius morphism of X. From this it follows that E 0 is given by a representation of the fundamental group-scheme of X in G.

Coding Theory

The workshop on Coding Theory has brought together leading researchers in several key areas of mathematical coding theory. On the side of many mathematicians there were computer scientist and electrical engineers present. Participants came from many countries and the group included both senior and junior researchers. Ever since its conception in the late 1940's, the theory of error-correcting codes has established itself as one of the central areas in mathematics. Coding theory lies naturally at the intersection of a large number of disciplines in pure and applied mathematics: algebra and number theory, probability theory and statistics, communication theory, discrete mathematics and combinatorics, complexity theory, and statistical physics, are just but a few areas which have brought about very interesting applications in coding theory in recent years. The multitude of methods and means to construct and analyze codes and their properties suggests that a workshop with the explicit aim of bringing together researchers in different sub-fields of coding theory is necessary for cross-fertilization of ideas and global advancement of the field. The following topics were covered during the workshop. Combinatorial and probabilistic coding theory: This area has experienced a huge revival in recent years because of its success in the design of codes with superior performance. Very roughly, in this area combinatorial structures are used to construct error-correcting codes, and properties of these structures are used to design and analyze efficient encoding and decoding algorithms for the codes. One of the most prominent examples in this area is furnished by the class of LDPC codes. These codes are constructed from sparse bipartite graphs. More generally Michael Tanner showed in the 80's how to construct ‘general codes on graphs’. The sparsity of the graph provides methods for construction of low complexity encoders and decoders. The graphs need to be designed in such a way as to facilitate an optimal operation of the algorithms. To achieve this goal researchers have developed and applied methods from probability theory and statistics, algebra, discrete mathematics, number theory, and statistical physics. Algebraic coding theory: Algebraic coding theory primarily investigates codes obtained from algebraic constructions. Prime examples of this area of coding theory are codes from algebraic geometry, and codes obtained from algebraically constructed expander graphs. This discipline is almost as old as the coding theory itself, and has attracted (and continues to attract) some of the brightest minds in the field. Among the most exciting advances in this field in recent years has been the invention of list-decoding algorithms for various classes of algebraic codes. Such decoding algorithms yield for a received word a short list of codewords that have at most a given distance \tau to the received word. The size of the list depends on the distance \tau . The methods in this field are mostly algebraic and make use of various properties of multivariate polynomials, or more generally, the properties of “well-behaved” functions in the function field of an irreducible variety. Methods from algebraic geometry are very important in this area. On the computational side the field naturally embedds in the theory of Gröbner bases. There are emerging relationships between this area and codes on graphs, the leading question being whether or not it is possible to match the superior performance of graph-based codes with list-decoding algorithms, or at least with algorithms that are derived from list-decoding algorithms. Theoretical computer science: Theoretical computer science has cotributed a large number of ideas to coding theory. The above mentioned analysis and design of LDPC codes, and the conception of list-decoding algorithms are two prime examples of such contributions. The reader will find it interesting to study in more details the summary of the talks collected in this report.

The $${\overline{\partial}}$$ -equation on homogeneous varieties with an isolated singularity

Let X be a regular irreducible variety in CP n 1 , Y the associated homogeneous variety in C n , and N the restriction of the universal bundle of CP n 1 to X. In the present paper, we compute the obstructions to solving the @-equation in the L p -sense on Y for 1 p1 in terms of cohomology groups H q (X;O(N )). That allows to identify obstructions explicitly if X is specied more precisely, for example if it is equivalent to CP 1 or an elliptic curve.

A trace formula for rigid varieties, and motivic Weil generating series for formal schemes

We establish a trace formula for rigid varieties X over a complete discretely valued field, which relates the set of unramified points on X to the Galois action on its étale cohomology. Next, we show that the analytic Milnor fiber of a morphism f at a point x completely determines the formal germ of f at x. We develop a theory of motivic integration for formal schemes of pseudo-finite type over a complete discrete valuation ring R, and we introduce the Weil generating series of a regular formal R-scheme $${\mathfrak{X}}$$ of pseudo-finite type, via the construction of a Gelfand-Leray form on its generic fiber. When $${\mathfrak{X}}$$ is the formal completion of a morphism f from a smooth irreducible variety to the affine line, then its Weil generating series coincides (modulo normalization) with the motivic zeta function of f. When $${\mathfrak{X}}$$ is the formal completion of f at a closed point x of the special fiber $${f^{-1}(0)}$$ , we obtain the local motivic zeta function of f at x.

A degree bound for globally generated vector bundles

Let E be a globally generated vector bundle of rank e ≥ 2 over a reduced irreducible projective variety X of dimension n defined over an algebraically closed field of characteristic zero. Let L := det(E). If deg(E) := deg(L) = L n > 0 and E is not isomorphic to $${\mathcal{O}_X^{e-1}\oplus L}$$ , we obtain a sharp bound $${\rm deg}(E)\geq h^0(X,E)-e$$ on the degree of E, proving also that deg(L) = h 0(X, L) − n if equality holds. As an application, we obtain a Del Pezzo–Bertini type theorem on varieties of minimal degree for subvarieties of Grassmannians, as well as a bound on the sectional genus for subvarieties $${X \subset \mathbb{G}(k,N)}$$ of degree at most N + 1.

On the Moduli Space of Pointed Algebraic Curves of Low Genus II ---Rationality---

We show that the moduli space $\mathcal{M}_{g,1}^N$ of pointed algebraic curves of genus $g$ with a given numerical semigroup $N$ is an irreducible rational variety if $N$ is generated by less than five elements for low genus ($ g \leq 6$) except one case. As a corollary to this result, we get a computational proof of the rationality of the moduli space $\mathcal{M}_{g,1}$ of pointed algebraic curves of genus $g$ for $1 \leq g \leq 3$. If $g \leq 5$, we also have that $\mathcal{M}_{g,1}^N$ is an irreducible rational variety for any semigroup $N$ except two cases. It is known that such a moduli space $\mathcal{M}_{g,1}^N$ is non-empty for $g \leq 7$.

Finite Generation of the Group of Eigenvalues for Sets of Derivations or Automorphisms of Division Algebras

Let D be a division algebra such that D⊗ D o is a Noetherian algebra, and Δ be a set of derivations or automorphisms of the division algebra D, then the group Ev(Δ) of common eigenvalues (i.e., weights) is a finitely generated abelian group. Typical examples of D are the quotient division algebra Frac (𝒟 (X)) of the ring of differential operators 𝒟 (X) on a smooth irreducible affine variety X over a field K of characteristic zero, and the quotient division algebra Frac (U (𝔤)) of the universal enveloping algebra U(𝔤) of a finite dimensional Lie algebra 𝔤.