Components of the Hilbert scheme of smooth space curves with the expected number of moduli

For any d and g such that the Brill-Noether number (d,g,3) is negative, d20 and gf(d), where f(d)=d^3/2/(6·2^1/2) + lower order terms, there exists a regular component of H _d,g ³ (the closure in Hilb₃ of the open set parametrizing smooth, connected, non-degenerate curves in ₃) with the expected number of moduli. Moreover, examples of the fact that such components are not unique are given.     

On the irreducibility of the Hilbert scheme of space curves

Denote by the Hilbert scheme parametrizing smooth irreducible complex curves of degree and genus embedded in . In 1921 Severi claimed that is irreducible if . As it has turned out in recent years, the conjecture is true for and , while for it is incorrect. We prove that , and are irreducible, provided that , and , correspondingly. This augments the results obtained previously by Ein (1986), (1987) and by Keem and Kim (1992).

     

On the Hilbert scheme of curves in higher-dimensional projective space

In this paper we prove that, for anyn≥3, there exist infinitely manyr∈N and for each of them a smooth, connected curveC _r in ℙ ^r such thatC _r lies on exactlyn irreducible components of the Hilbert scheme Hilb(ℙ ^r ). This is proven by reducing the problem to an analogous statement for the moduli of surfaces of general type.     

Stratification of the Hilbert scheme of space curves with a stable normal bundle

Ford≥3g and 1≤s≤[g/2], we study the strataN _{d, g(s)} of degreed genusg spaces curvesC whose normal bundleN _C is stable with stability degree (integer of Lange-Narasimhan) σ(N _C)=2s. We prove thatN _{d, g(s)} has an irreducible component of the right dimension whose general curve has a normal bundle with the right number of maximal subbundles. We consider also the semi-stable case (s=0), obtaining similar results. We prove our results by studying the normal bundles of reducible curves and their deformations. Both authors were partially supported by MIUR and GNSAGA of INdAM (Italy).     

A remark on hyperelliptic curves

Partially supported by IHES and NSF.     

A remark on regularization in Hilbert spaces

Every graph with uncountable chromatic number contains for every finiten ann-connected subgraph with infinite degrees which has uncountable chromatic number.     

The Hilbert function of curves on certain smooth quartic surfaces

The Riemann-Roch problem for divisors on a smooth surface in ³ is studied. This problem is solved for some smooth quartic surfaces which are called Mori quartics; as a consequence the Hilbert function of any integral curve on a Mori quartic is determined.     

Computing genus 2 curves from invariants on the Hilbert moduli space

We give a new method for generating genus 2 curves over a finite field with a given number of points on the Jacobian of the curve. We define two new invariants for genus 2 curves as values of modular functions on the Hilbert moduli space and show how to compute them. We relate them to the usual three Igusa invariants on the Siegel moduli space and give an algorithm to construct curves using these new invariants. Our approach simplifies the complex analytic method for computing genus 2 curves for cryptography and reduces the amount of computation required.     

The play operator on the rectifiable curves in a Hilbert space

The vector play operator is the solution operator of a class of evolution variational inequalities arising in continuum mechanics. For regular data, the existence of solutions is easily obtained from general results on maximal monotone operators. If the datum is a continuous function of bounded variation, then the existence of a weak solution is usually proved by means of a time discretization procedure. In this paper we give a short proof of the existence of the play operator on rectifiable curves making use of basic facts of measure theory. We also drop the separability assumptions usually made by other authors. Copyright © 2007 John Wiley & Sons, Ltd.     

A moduli space of non-compact curves on a complex surface

We show that under mild boundary conditions the moduli space of non-compact curves on a complex surface is (locally) an analytic subset of a ball in a Banach manifold, defined by finitely many holomorphic functions.