arXiv subjects
Publications and source records attributed to Ziv Ran.
We study the relative Hilbert scheme of a family of nodal (or smooth) curves via its (birational) cycle map, going to the relative symmetric product. We show the cycle map is the blowing up of the discriminant locus, which consists of cycles with multiple points. We derive an intersection calculus for Chern classes of tautological bundles on the relative Hilbert scheme, which has applications to enumerative geometry.
We study completely the Hilbert scheme and punctual Hilbert scheme of a nodal curve, and the relative Hilbert scheme of a family of curves acquiring a node. The results are then extended, less completely, to flag Hilbert schemes, parametrizing chains of subschemes. We find, notably, that if the total space $X$ of a family $X/B$ is smooth (over an algebraically closed field $\k$), then the relative Hilbert scheme $Hilb_m(X/B)$ is smooth over $\k$ and the flag Hilbert schemes are normal and locally complete intersection, but generally singular.
We determine the splitting (isomorphism) type of the normal bundle of a generic genus-0 curve with 1 or 2 components in any projective space, as well as the (sometimes nontrivial) way the bundle deforms locally with a general deformation of the curve. We deduce an enumerative formula for divisorial loci of smooth rational curves whose normal bundle is of non-generic splitting type.
Given a family $X/B$ of nodal curves, we construct canonically and compatibly with base-change, via an explicit blow-up of the Cartesian product $X^r/B$, a family $W^r(X/B)$ parametrizing length-$r$ subschemes of fibres of $X/B$ (plus some additional data). Though $W^r(X/B)$ is singular, the important sheaves on it are locally free, which allows us to study intersection theory on it and deduce enumerative applications, including some relative multiple point formulae, enumerating the length-$r$ schemes contained simultaneously in some fibre of $X/B$ and some fibre of a given map from $X$ to a smooth variety.
This revision contains some additional corrections and references, including a reference to Zeuthen suggested by Kleiman. For possible subsequent revisions, check http://math.ucr.edu/~ziv/papers/1nodal.pdf
We prove some lower bounds on certain nonegative twists of the canonical bundle of a subvariety of a generic hypersurface in projective space. In particular we prove that the generic sextic threefold contains no rational or elliptic curves and no nondegenerate curves of genus 2.
We extend the results of the eprint math.AG/0204256, obtained with Herb Clemens, about sections of nonnegative twists of the canonical bundle of subvarieties of generic hypersurfaces, to the case of the (-1) twist.
Let $Y \subset ¶^r$ be a normal nondegenerate m-dimensional subvariety and let $σ(Y)$ denote the maximum dimension of a subvariety $Z \subset Y_{smooth}$ such that $Z$ contains a generic point of some divisor on $Y$ and the tangent planes $T_y Y$ for all $y \in Z$ are contained in a fixed hyperplane. In this article we study the double locus $D \subset $Y$ of its generic projection to $¶^{r-1}$, proving that if the secant variety of $Y$ is the whole space and $σ(Y) < 2m - r + 1$, then $D$ is irreducible. Applying Zak's Tangency theorem we deduce the irreducibility of $D$ when $m > 2(r-1)/3$. The latter implies a version of Zak's Linear Normality theorem.
The main aim of this paper is to develop general algebraic and cohomological tools for the study of the local geometry of moduli and parameter spaces in Algebraic Geometry, culminating in the so-called Hitchin (or KZ) (projective) connection over the Moduli of curves and the proof of its flatness. Among other things, we notably give a 'modular' (i.e. in terms of the modular problem) construction of the differential operators- of any order- acting on (general)modular vector bundles, and in particular for the Lie algebra of vector fields on a (general) Moduli space, with its natural action on functions. The methods involve mostly Lie algebras and their cohomology.
We prove some lower bounds on certain twists of the canonical bundle of a codimension-2 subvariety of a generic hypersurface in projective space. In particular we prove that the generic sextic threefold contains no rational or elliptic curves and no nondegenerate curves of genus 2.
For a semistable reflexive sheaf $E$ of rank $r$ and $c_1=a$ on $¶^n$ and an integer $d$ such that $r|ad$, we give sufficient conditions so that the restriction of $E$ on a generic rational curve of degree $d$ is balanced, i.e. a twist of the trivial bundle (for instance, if $E$ has balanced restriction on a generic line, or $r=2$ or $E$ is an exterior power of the tangent bundle). Assuming this, we give a formula for the 'virtual degree', interpreted enumeratively, of the locus of rational curves of degree $d$ on which the restriction of $E$ is not balanced, generalizing a classical formula due to Barth for the degree of the divisor of jumping lines of a semistable rank-2 bundle.
We give some bounds on the anticanonical degrees of Fano varieties with Picard number 1 and mild singularities, extending results of Kollár et al. from the early 90's and improving them even in the smooth case. The proof is based on a study of positivity properties of sheaves of differential operators on ample line bundles, and avoids the use of rational curves and bend-and-break. This note is a self-contained exposition of the main ideas of math.AG/9811022
We consider normal projective n-dimensional varieties X whose anticanonical divisor class -K is ample and where every Weil divisor is a rational multiple of K. The index i is the largest integer such that K/i exists as a Weil divisor. We show (i) if X has log-terminal singularities, and in addition 1-forms on the smooth part of X are holomorphic on a resolution, then (-K)^n =< (max(in,n+1))^n; (ii) if the tangent sheaf of X is semistable, then (-K)^n =<(2n)^n. The proof is based on some elementary but possibly surprising slope estimates on sheaves of differential operators on plurianticanonical sheaves. Unlike previous arguments in the smooth case (Nadel, Campana, Kollar-Miyaoka-Mori), rational curves and rational connectedness are not used.
This paper is a sequel to math.AG/9810041 (whose abstract should have mentioned the fact that a version of the jacobi complex and higher-order Kodaira-Spencer maps were also discovered independently by Esnault and Viehweg). We give a canonical algebraic construction for the variation of Hodge structure associated to the universal m-th order deformation of a compact Kahler manifold without vector fields. Specializing to the case of a Calabi-Yau manifold, we give a formula for the mth derivative of its period map and deduce formal defining equations for the image (Schottky relations).
This paper gives a canonical construction, in terms of additive cohomological functors, of the universal formal deformation of a compact complex manifold without vector fields (more generally of a faithful $g$-module, where $g$ is a sheaf of Lie algebras without sections). The construction is based on a certain (multivariate) Jacobi complex $J(g)$ associatd to $g$: indeed ${\mathbb C}\oplus {\mathbb H}^0(J(g))^*$ is precisely the base ring of the universal deformation.
Extending the method of Part I (alg-geom/9704004), we give recursive formulae for: the genus-1 Severi degree (formula first found by Getzler), the degree of the variety of 1-cuspidal curves of genus 0 or 1, and the linear (sectional) geometric genus of the genus-0 and genus-1 Severi varieties. The proof is short and elementary.
We give a 'recursive' formula (in terms of reducible limits) for counting rational curves on a variety moving in any sufficiently large and well-behaved family. Our approach is completely elementary and makes no use of moduli spaces for maps or the like.
We describe a method for counting maps of curves of given genus (and variable moduli) to $\Bbb P^2$, essentially by splitting the $\Bbb P^2$ in two; then specialising to the case of genus 0 we show that the method of quantum cohomology may be viewed as the 'mirror' of the former method where one splits the $\Bbb P^1$ rather than the $\Bbb P^2$, and we indicate a proof of the associativity of quantum multiplication based on this idea.