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At least 91 records · Page 5Linked to original sources
arXiv1994

Computing Multidimensional Residues

Given n polynomials in n variables with a finite number of complex roots, for any of their roots there is a local residue operator assigning a complex number to any polynomial. This is an algebraic, but generally not rational, function of the coefficients. On the other hand, the global residue, which is the sum of the local residues over all roots, depends rationally on the coefficients. This paper deals with symbolic algorithms for evaluating that rational function. Under the assumption that the deformation to the initial forms is flat, for some choice of weights on the variables, we express the global residue as a single residue integral with respect to the initial forms. When the input equations are a Groebner basis, this leads to an efficient series expansion algorithm for global residues, and to a vanishing theorem with respect to the corresponding cone in the Groebner fan. The global residue of a polynomial equals the highest coefficient of its (Groebner basis) normal form, and, conversely, the entire normal form is expressed in terms of global residues. This yields a method for evaluating traces over zero-dimensional complete intersections. Applications include the counting of real roots, the computation of the degree of a polynomial map, and the evaluation of multivariate symmetric functions. All algorithms are illustrated for an explicit system in three variables.

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arXiv1994

Conformal blocks, fusion rules and the Verlinde formula

The Verlinde formula computes the dimension of certain vector spaces ("conformal blocks") associated to a Rational Conformal Field Theory. In this paper we show how this can be made rigorous for one particular such theory, the WZW model. Thanks to the results of [B-L], [F] and [T-U-Y], this gives the dimension of the space of global sections of the determinant line bundles (and its multiples) on the moduli space of vector bundles with fixed rank and determinant.

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arXiv1994

An Algorithmic Proof of Suslin's Stability Theorem over Polynomial Rings

Let $k$ be a field. Then Gaussian elimination over $k$ and the Euclidean division algorithm for the univariate polynomial ring $k[x]$ allow us to write any matrix in $SL_n(k)$ or $SL_n(k[x])$, $n\geq 2$, as a product of elementary matrices. Suslin's stability theorem states that the same is true for the multivariate polynomial ring $SL_n(k[x_1,\ldots ,x_m])$ with $n\geq 3$. As Gaussian elimination gives us an algorithmic way of finding an explicit factorization of the given matrix into elementary matrices over a field, we develop a similar algorithm over polynomial rings.

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arXiv1994

Geometric invariant theory and flips

We study the dependence of geometric invariant theory quotients on the choice of a linearization. We show that, in good cases, two such quotients are related by a flip in the sense of Mori, and explain the relationship with the minimal model programme. Moreover, we express the flip as the blow-up and blow-down of specific ideal sheaves, leading, under certain hypotheses, to a quite explicit description of the flip. We apply these ideas to various familiar moduli problems, recovering results of Kirwan, Boden-Hu, Bertram-Daskalopoulos- Wentworth, and the author. Along the way we display a chamber structure, following Duistermaat-Heckman, on the space of all linearizations. We also give a new, easy proof of the Bialynicki-Birula decomposition theorem.

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arXiv1994

Local structure of the moduli space of vector bundles over curves

We analyze the local structure of the moduli space of semi-stable bundles on a curve. In particular, a complete description of the local structure is given in the rank 2 case. We obtain as a corollary of this analysis new results about the Kummer variety and the Coble quartic associated to a canonical genus 3 curve.

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arXiv1994

Spin canonical invariants of 4-manifolds and algebraic surfaces

The paper is a colloquial-style discussion of invariants of algebraic surfaces analogous to the Donaldson polynomials, arising from moduli spaces of ``jumping'' Yang--Mills instantons, or moduli spaces of jumping vector bundles. The invariants have the following applications: (1) to the Van de Ven conjecture that the Kodaira dimension is a diffeomorphism invariant; (2) to proving that algebraic surfaces with $p_g > 0$ have a proper sublattice of $H^2(X,\Z)$ invariant under diffeomorphism; (3) to proving the same result as (2) for surfaces with $p_g = 0$, in particular the Barlow surface.

alg-geom↗
arXiv1994

Universal Hodge Bundle and Mumford Isomorphisms on the Abelian Inductive Limit of Teichmüller Spaces

Let $\Mg$ denote the moduli space of compact Riemann surfaces of genus $g$. Mumford had proved that, for each fixed genus $g$, there are isomorphisms asserting that certain higher $DET$ bundles over $\Mg$ are certain fixed (genus-independent) tensor powers of the Hodge line bundle on $\Mg$. Here we obtain a coherent, genus-independent description of the Mumford isomorphisms over certain infinite-dimensional ``universal'' parameter spaces of compact Riemann surfaces (with or without marked points). We work with an inductive limit of Teichmüller spaces comprising complex structures on a certain ``solenoidal Riemann surface'', $H_{\infty,abel}$. The main result constructs these universal $DET$ and Hodge line bundles on this direct limit, equips them with their Quillen metrics, and proves that they are related by the appropriate Mumford isomorphisms. Our work can be viewed as a contribution towards a non-perturbative formulation of the bosonic string theory, since it provides an interpretation of the Polyakov measure in a genus-independent fashion.

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arXiv1994

Quadrature Mirror Filters and Loop Groups

In this paper we want to show, that the finite impulse response quadratic mirror filters (QMF) associated to a tower of grids $Γ\subset H=\bf Z^n$ can be identified with a right coset of the subgroup Fix$(T_{Γ^{\perp}}$,Map$ ({\bf T}^n\to U(N)$: poly) of the group of polynomial loops Map$({\bf T}^n\to U(N)$: poly) with $N=|H/Γ|$. The QMF with some vanishing moments can be identified with cosets of subgroups. The problem to parameterize all finite impulse response QMF in arbitrary space dimensions is now equivalent to factorize all polynomial loops.

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arXiv1994

Canonical Generators for the Cohomology of Moduli of Parabolic Bundles on Curves

We determine generators of the rational cohomology algebras of moduli spaces of parabolic vector bundles on a curve, under some `primality' conditions on the parabolic datum. These generators are canonical in a precise sense. Our results are new even for usual vector bundles (i.e., vector bundles without parabolic structure) whose rank is greater than 2 and is coprime to the degree; in this case, they are generalizations of a theorem of Newstead on the moduli of vector bundles of rank 2 and odd degree.

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arXiv1994

Semistable sheaves on Del Pezzo surfaces and Kronecker modules

In this paper we deal with semistable sheaves which can be represented as the cokernel of an injective (or as the kernel of a surjective) morphism $E_1\otimes\CC^m\longrightarrow E_2\otimes\CC^n$ , where $E_1$ and $E_2$ are exceptional bundles. To each such a sheaf we assign the linear map $\CC^m\otimes\hom^*(E_1,E_2)\longrightarrow \CC^n$. We obtain sufficient conditions on the topological invariants of the sheaves for the moduli space of the sheaves to be isomorphic to the quotient of an open subset in $\PP{\cal L}(\CC^m\otimes\hom^*(E_1,E_2),\CC^n)$ under the action of SL$(\CC^m)\times$SL$(\CC^n)$.

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arXiv2019

Classification of Varieties with Canonical Curve Section via Gaussian maps on Canonical Curves

Let $C \subset P^{g-1}$ be a smooth canonical curve of genus $g \geq 3$. The purpose of this article is to further develop a method to classify varieties having $C$ as their curve section, using Gaussian map computations. In a previous article a careful analysis of the degeneration to the cone over the hyperplane section was made for _prime_ Fano threefolds, that is Fano threefolds whose Picard group is generated by the hyperplane bundle. In this article we extend this method and classify Fano threefolds of higher index (which still have Picard number one). We are also able to classify Mukai varieties, i.e. varieties of dimension four or more with canonical curve sections.

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arXiv1994

Remarks on S. Lang's conjecture over function fields

In this short note, we will show the following weak evidence of S. Lang conjecture over function fields. Let f : X ---> Y be a projective and surjective morphism of algebraic varieties over an algebraically closed field k of characteristic zero, whose generic fiber is geometrically irreducible and of general type. If f is not birationally trivial, then there are countably many proper closed varieties { Z_i } of X such that every quasi-section of f is contained in the union of Z_i.

alg-geom↗
arXiv1994

Grothendieck duality via homotopy theory

Grothendieck proved that if $f:X\longrightarrow Y$ is a proper morphism of nice schemes, then $Rf_*$ has a right adjoint, which is given as tensor product with the relative canonical bundle. The original proof was by patching local data. Deligne proved the existence of the adjoint by a global argument, and Verdier showed that this global adjoint may be computed locally. In this article we show that the existence of the adjoint is an immediate consequence of Brown's representability theorem. It follows almost as immediately, by ``smashing'' arguments, that the adjoint is given by tensor product with a dualising complex. Verdier's base change theorem is an immediate consequence.

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arXiv1995

Spectral Covers

This is a survey of various results about spectral covers and their relationship to Higgs bundles. To a G-principal Higgs bundle on a variety S corresponds a cameral cover \widetilde{S} of S (a W-Galois cover, where W is the Weyl group of G) together with a sheaf on \widetilde{S} which in simple cases is a line bundle, and is W-equivariant up to certain twists and shifts. Various other types of spectral covers, depending on the choice of a representation or weight of G, arise as associated objects of \widetilde{S}. We focus on the decomposition of the Picards of these spectral covers into Pryms (this includes various well-known Prym identities as special cases) and on the interpretation, in the spirit of Hitchin's abelianization program, of a distinguished Prym component as parameter space for higgs bundles.

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arXiv1995

Algebraic Barth-Lefschetz theorems

Using results of Hironaka-Matsumura and Faltings, we prove a strong version of the well known Fulton-Hansen connectivity theorem for weighted projective spaces. As a consequence we get the following result. If $Y$ is an irreducible subvariety of the $n$-dimensional projective space (over a field of arbitrary characteristic), then the diagonal embedding $Δ_Y$ is $G_3$ in $Y\times Y$. This fact implies a generalized version (with a characteristic-free proof) of a result of Ogus (in char. zero) and Speiser (in positive characteristic).

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arXiv1995

Wall-crossing formulas, Bott residue formula and the Donaldson invariants of rational surfaces

We study the Donaldson invariants of rational surfaces and their dependence on the chambers in the ample cone. We build on a previous joint paper in which we have expressed the change of the Donaldson invariants on an algebraic surface $S$ under crossing a so-called good wall in terms of certain intersection numbers on Hilbert schemes of points on $S$. In the current paper we assume that $S$ is rational. Thish enables us to apply the Bott residue theorem to evaluate these intersection numbers with the help of a computer. Combining this with the blowup formulas we obtain an algorithm for computing the Donaldson invariants of all rational surfaces in almost all chambers. In particular we compute all the $SU(2)$- and $SO(3)$-Donaldson invariants of the projective plane of degree at most 50.

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