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At least 217 records · Page 12Linked to original sources
arXiv1996

The Eisenbud-Koh-Stillman Conjecture on Linear Syzygies

It is proved, as was conjectured by Eisenbud-Koh-Stillman, that for a finitely generated graded module $M$ over the symmetric algebra $S(V)$, if the Koszul group ${\cal K}_{p,0}(M,V)\ne 0$, then the set of rank 1 relations in $M_0\otimes V$ has dimension $\ge p$. The method is by using ``exterior minors" to study syzygies of an ideal derived from the Koszul class in the exterior algebra. As a consequence, a conjecture of Lazarsfeld and myself that a set $Z$ of $2r+1-p$ points in ${\bf P}^r$ for which property $N_p$ fails has a subset $Z'$ of at least $2{\rm dim}(L) +2-p$ points lying on a linear space $L$ such that property $N_p$ fails for $Z'$.

alg-geom↗
arXiv1996

Bounds on cohomology and Castelnuovo-Mumford regularity

The Castelnuovo-Mumford regularity reg(X) of a projective scheme X was introduced by Mumford by generalizing ideas of Castelnuovo. The interest in this concept stems partly from the fact that X is m-regular if and only if for every p \geq 0 the minimal generators of the p-th syzygy module of the defining ideal I of X occur in degree \leq m + p. There are some bounds in the case that X is a locally Cohen-Macaulay scheme. The aim of this paper is to extend and improve these results for so-called (k,r)-Buchsbaum schemes. In order to prove our theorems, we need to apply a spectral sequence. We conclude by describing two sharp examples and open problems.

alg-geom↗
arXiv1996

Transformation de Fourier generalisee

In this paper I construct a geometric transformation for generalized 1-motives which extends the Fourier-Mukai transformation for O-Modules on abelian varieties, the geometric Fourier transformation for D-Modules on vector spaces and the geometric Mellin transformation for D-Modules on tori. In particular, I construct an equivalence of triangulated categories between the derived category of quasi-coherent D-Modules on an abelian variety and the derived category of quasi-coherent O-Modules on the universal extension of the dual abelian variety. This equivalence has also been obtained by Mitchell Rothstein.

alg-geom↗
arXiv1996

Smoothness, Semistability, and Toroidal Geometry

We provide a new proof of the following result: Let $X$ be a variety of finite type over an algebraically closed field $k$ of characteristic 0, let $Z\subset X$ be a proper closed subset. There exists a modification $f:X_1 \rar X$, such that $X_1$ is a quasi-projective nonsingular variety and $Z_1 = f^{-1}(Z)_\red$ is a strict divisor of normal crossings. Needless to say, this theorem is a weak version of Hironaka's well known theorem on resolution of singularities. Our proof has the feature that it builds on two standard techniques of algebraic geometry: semistable reduction for curves, and toric geometry. Another proof of the same result was discovered independently by F. Bogomolov and T. Pantev. The two proofs are similar in spirit but quite different in detail.

alg-geom↗
arXiv1996

Weak Hironaka theorem

The purpose of this note is to give a simple proof of the following theorem: Let $X$ be a normal projective variety over an algebraically closed field $k$, $\op{char} k = 0$ and let $D \subset X$ be a proper closed subvariety of $X$. Then there exist a smooth projective variety $M$, a strict normal crossings divisor $R \subset M$ and a birational morphism $f : M \to X$ with $f^{-1} D = R$. The method of proof is inspired by A.J. de Jong alteration ideas. We also use a multidimensional version of G.Belyi argument which allows us to simplify the shape of a ramification divisor. By induction on the dimension of $X$ the problem is reduced to resolving toroidal singularities. This process however is too crude and does not permit any control over the structure of the birational map $f$. A different proof of the same theorem was found independently by D. Abramovich and A.J. de Jong. The approach is similar in both proofs but they seem to be rather different in details.

alg-geom↗
arXiv1996

Higher dimensional examples of manifolds whose adjoint bundles are not spanned

Let $(X,L)$ be an $n$-dimensional polarized variety. Fujita's conjecture says that if $L^n>1$ then the adjoint bundle $K_X+nL$ is spanned and $K_X+(n+1)L$ is very ample. There are some examples such that $K_X+nL$ is not spanned or $K_X+(n+1)L$ is not very ample. These are $(¶^n,Ø(1))$, hypersurface $M$ of degree $6$ in weighted projective space $¶(3,2,1,1,\cdots ,1)$ with $Ø_M(1)$ and numerically Godeaux surface etc. Numerically Godeaux surface is the quotient space of a Fermat type hypersurface of degree $5$ in $¶^3$ by an action of order $5$. These examples are not so much. We construct new examples for any dimention.

alg-geom↗
arXiv1996

d-gonality of modular curves and bounding torsions

We study the problem of $d$-gonality of the modular curve $X_0(N)$. As a result, we can give an upperbound of the level $N$ by means of $d$. This generalizes Ogg's result on hyperelliptic modular curves ($d = 2$). As a corollary of this result, we prove an analogue of the strong Uniform Boundedness Conjecture for elliptic curves defined over the function fields of curves. If a base curve is $d$-gonal, we can bound orders of torsions of Mordell-Weil groups in terms of $d$ uniformly.

alg-geom↗
arXiv1996

Threefolds in $\Bbb P^5$ with a 3-dimensional family of plane curves

A classification theorem is given of smooth threefolds of $\Bbb P^5$ covered by a family of dimension at least three of plane integral curves of degree $d\geq 2.$ It is shown that for such a threefold $X$ there are two possibilities: \item{(1)} $X$ is any threefold contained in a hyperquadric; \item{(2)} $d\leq 3$ and $X$ is either the Bordiga or the Palatini scroll.

alg-geom↗
arXiv2013

On top Chern classes of universal bundles on moduli spaces of rank two coherent sheaves on the projective plane, or How to calculate the correlation function in SYM N=2 Nf=4 quantum field theory on complex projective plane

We explain how to calculate the correlation function for SYM N=2 SO(3)-gauge QFT with 4 flavors in terms of top Chern classes of the universal bundles over the moduli spaces of rank 2 stable torsion free coherent sheaves with det=-1 on the complex projective plane. We give a direct geometrical description of these moduli spaces, study a 2-dimensional torus action on them, and calculate several initial coefficients of the correlation function via Bott residue formula.

alg-geom↗
arXiv2009

Equivalences of derived categories and K3 surfaces

We consider derived categories of coherent sheaves on smooth projective varieties. We prove that any equivalence between them can be represented by an object on the product. Using this, we give a necessary and sufficient condition for equivalence of derived categories of two K3 surfaces.

alg-geom↗