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At least 253 records · Page 14Linked to original sources
arXiv1996

Log canonical singularities and complete moduli of stable pairs

1) Assuming log Minimal Model Conjecture, we give a construction of a complete moduli space of stable log pairs of arbitrary dimension generalizing directly the space M_{g,n} of pointed stable curves. Each stable pair has semi log canonical singularities. 2) We prove that a stable quasiabelian pair, defined by author and I.Nakamura as the limit of abelian varieties with theta divisors, has semi log canonical singularities.

alg-geom↗
arXiv1996

Complete intersections and rational equivalence

Two cycles on a projective variety over an algebraically closed field are shown to be rationally equivalent if and only if their difference equals a difference of complete intersections of a certain kind. Some of Bloch's conjectures for zero-cycles on surfaces can be restated more geometrically, which leads to several questions.

alg-geom↗
arXiv1996

On the Thomae formula for $Z_N$ curves

We shall give an elementary and rigorous proof of the Thomae formula for ${\bf Z}_N$ curves which was discovered by Bershadsky and Radul. Instead of using the determinant of the Laplacian we use the traditional variational method which goes back to Riemann, Thomae, Fuchs. In the proof we made explicit the algebraic expression of the chiral Szegö kernels and proves the vanishing of zero values of derivatives of theta functions with ${\bf Z}_N$ invariant $1/2N$ characteristics.

alg-geom↗
arXiv1996

A finiteness theorem for low-codimensional nonsingular subvarieties of quadrics

We prove that there are only finitely many families of codimension two nonsingular subvarieties of quadrics $\Q{n}$ which are not of general type, for $n=5$ and $n\geq 7$. We prove a similar statement also for the case of higher codimension. The case $n=6$ has been recently settled by Fania-Ottaviani. Keywords: Codimension two, Grassmannians, Lifting, Low codimension, Not of General Type, Polynomial Bound, Quadrics

alg-geom↗
arXiv1996

Counting plane curves of any genus

We obtain a recursive formula answering the following question: How many irreducible, plane curves of degree d and (geometric) genus g pass through 3d-1+g general points in the plane? The formula is proved by studying suitable degenerations of plane curves.

alg-geom↗
arXiv1997

Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface

We prove formulas (found by Witten in 1992 using physical methods) for intersection pairings in the cohomology of the moduli space M(n,d) of stable holomorphic vector bundles of rank n and degree d (assumed coprime) on a Riemann surface of genus g greater than or equal to 2. We also use these formulas for intersection numbers to obtain a proof of the Verlinde formula for the dimension of the space of holomorphic sections of a line bundle over M(n,d).

alg-geom↗
arXiv1996

Monodromy of hypergeometric functions arising from arrangements of hyperplanes

Given an arrangement of hyperplanes in $¶^n$, possibly with non-normal crossings, we give a vanishing lemma for the cohomology of the sheaf of $q$-forms with logarithmic poles along our arrangement. We give a basis for the ideal $\cal J$ of relations for the Orlik-Solomon's algebra. Under certain genericity conditions it was shown by H.~Esnault, V.~Schechtman and E.~Viehweg that the cohomology of a local system is given by the Aomoto complex. We generalize this result to a deformation of local systems obtained via a deformation of our arrangement. We calculate the Gauß-Manin connection for this case. We give a basis for the Gauß-Manin bundle for which, with help of the basis for $\cal J$, we give then a method to calculate a representation of this connection. From here, with the results of K-T.~Chen or P.~Deligne, one can calculate the monodromy representation. This gives a generalization of the hypergeometric functions.

alg-geom↗
arXiv1997

Estimates of the number of rational mappings from a fixed variety to varieties of general type

First we find effective bounds for the number of dominant rational maps $f:X \rightarrow Y$ between two fixed smooth projective varieties with ample canonical bundles. The bounds are of the type $\{A \cdot K_X^n\}^{\{B \cdot K_X^n\}^2}$, where $n=dimX$, $K_X$ is the canonical bundle of $X$ and $A,B $ are some constants, depending only on $n$. Then we show that for any variety $X$ there exist numbers $c(X)$ and $C(X)$ with the following properties: For any threefold $Y$ of general type the number of dominant rational maps $f:X \r Y$ is bounded above by $c(X)$. The number of threefolds $Y$, modulo birational equivalence, for which there exist dominant rational maps $f:X \r Y$, is bounded above by $C(X)$. If, moreover, $X$ is a threefold of general type, we prove that $c(X)$ and $C(X)$ only depend on the index $r_{X_c}$ of the canonical model $X_c$ of $X$ and on $K_{X_c}^3$.

alg-geom↗
arXiv1996

Density and completeness of subvarieties of moduli spaces of curves or abelian varieties

Let $V$ be a subvariety of codimension $\leq g$ of the moduli space $\cA_g$ of principally polarized abelian varieties of dimension $g$ or of the moduli space $\tM_g$ of curves of compact type of genus $g$. We prove that the set $E_1(V)$ of elements of $V$ which map onto an elliptic curve is analytically dense in $V$. From this we deduce that if $V \subset \cA_g$ is complete, then $V$ has codimension equal to $g$ and the set of elements of $V$ isogenous to a product of $g$ elliptic curves is countable and analytically dense in $V$. We also prove a technical property of the conormal sheaf of $V$ if $V \subset \tM_g$ (or $\cA_g$) is complete of codimension $g$.

alg-geom↗
arXiv1996

Counting rational curves with multiple points and Gromov-Witten invariants of blow-ups

We study Gromov-Witten invariants on the blow-up of P^n at a point, which is probably the simplest example of a variety whose moduli spaces of stable maps do not have the expected dimension. It is shown that many of these invariants can be interpreted geometrically on P^n as certain numbers of rational curves having a multiple point of given order at the blown up point. Moreover, all these invariants can actually be calculated, giving enumerative invariants of P^n which have not been known before.

alg-geom↗
arXiv1996

Equivariant resolution of singularities in characteristic 0

A new proof of equivariant resolution of singularities under a finite group action in characteristic 0 is provided. We assume we know how to resolve singularities without group action. We first prove equivariant resolution of toroidal singularities. Then we reduce the general case to the toroidal case.

alg-geom↗
arXiv1996

Mathematical Aspects of Mirror Symmetry

Lecture notes from 1993 Park City lectures and 1994 Trento lectures. The focus of these lectures is on giving a mathematical description of the A-model and B-model correlation functions on a Calabi--Yau manifold, and a precise mathematical conjecture relating these for a pair of mirror manifolds.

alg-geom↗
arXiv1996

Crepant Terminalisations and Orbifold Euler Numbers for SL(4) Singularities

Let $X$ and $Y$ be two analytic canonical Gorenstein orbifolds. A resolution of singularities $Y\to X$ is called an Euler resolution if $Y$ and $X$ have the same orbifold Euler number. If $Y$ is only terminal rather than smooth, it is called an Euler terminalisation. It is proved that Euler terminalisations exist for toric varieties in any dimension, for 4-dimensional toroidal varieties, and for singularities $\C^4/G$ where $G$ belongs to certain classes of $\SL(4)$ subgroups. The method of proof is expected to be applicable to a sizeable number of finite $\SL(4)$ subgroups and to lead to a generalisation of the Dixon-Harvey-Vafa-Witten orbifold Euler number conjecture to dimension~4.

alg-geom↗
arXiv1996

Partial Resolutions of Orbifold Singularities via Moduli Spaces of HYM-type Bundles

Let $Γ$ be a finite group acting linearly on $\C^n$, freely outside the origin, and let $N$ be the number of conjugacy classes of $Γ$ minus one. A construction of Kronheimer of moduli spaces $X_ζ$ of translation-invariant $Γ$-equivariant instantons on $\C^2$ is generalised to $\C^n$. The moduli spaces $X_ζ$ depend on a parameter $ζ\in\Q^N$. The following results are proved: for $ζ=0$, $X_0$ is isomorphic to $\C^n/Γ$; if $ζ\neq 0$, the natural maps $X_ζ\to X_0$ are partial resolutions. The moduli $X_ζ$ are furthermore shown to admit Kähler metrics which are Asymptotically Locally Euclidean (ALE). A description of the singularities of $X_ζ$ using deformation complexes is given, and is applied in particular to the case $Γ\subset\SU(3)$. It is conjectured that for general $Γ$ and generic $ζ$ that the singularities of $X_ζ$ are at most quadratic. When $Γ\subset\SU(3)$ a natural holomorphic 3-form is constructed on the smooth locus of $X_ζ$, which is conjectured to be non-vanishing. The morphims $X_ζ\to X_0$ are expected to be crepant resolutions and $X_ζ$ to be smooth for generic choices of the parameter $ζ$. Related open problems in higher-dimensional complex geometry are also mentioned. The paper has a companion paper which identifies the moduli $X_ζ$ with representation moduli of McKay quivers, and describes them completely in the case of abelian groups.

alg-geom↗
arXiv1996

Monomial Resolutions

Call a monomial ideal M "generic" if no variable appears with the same nonzero exponent in two distinct monomial generators. Using a convex polytope first studied by Scarf, we obtain a minimal free resolution of M. Any monomial ideal M can be made generic by deformation of its generating exponents. Thus, the above construction yields a (usually nonminimal) resolution of M for arbitrary monomial ideals, bounding the Betti numbers of M in terms of the Upper Bound Theorem for Convex Polytopes. We show that our resolutions are DG-algebras, and consider realizability questions and irreducible decompositions.

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