On the moduli of surfaces admitting genus two fibrations over elliptic curves (original) (raw)

On the genus of elliptic fibrations

Proceedings of the American Mathematical Society

A simply connected topological space is called elliptic if both π ∗ ( X , Q ) \pi _*(X, \mathbb {Q}) and H ∗ ( X , Q ) H^*(X, \mathbb {Q}) are finite-dimensional Q \mathbb {Q} -vector spaces. In this paper, we consider fibrations for which the fibre X X is elliptic and H ∗ ( X , Q ) H^*(X, \mathbb {Q}) is evenly graded. We show that in the generic cases, the genus of such a fibration is completely determined by generalized Chern classes of the fibration.

Moduli of Bundles over Rational Surfaces and Elliptic Curves II: Nonsimply Laced Cases

International Mathematics Research Notices, 2009

It is well-known that del Pezzo surfaces of degree 9 − n one-toone correspond to flat En bundles over an elliptic curve. In this paper, we construct ADE bundles over a broader class of rational surfaces which we call ADE surfaces, and extend the above correspondence to all flat G bundles over an elliptic curve, where G is any simply laced, simple, compact and simplyconnected Lie group. In the sequel, we will construct G bundles for nonsimply laced Lie group G over these rational surfaces, and extend the above correspondence to non-simply laced cases.

On rational maps between moduli spaces of curves and of vector bundles

arXiv: Algebraic Geometry, 2009

Let SU_C(2) be the moduli space of rank 2 semistable vector bundles with trivial de terminant on a smooth complex algebraic curve C of genus g > 1, we assume C non-hyperellptic if g > 2. In this paper we construct large families of pointed rational normal curves over certain linear sections of SU_C(2). This allows us to give an interpretation of these subvarieties of SUC(2) in terms of the moduli space of curves M_{0,2g}. In fact, there exists a natural linear map SU_C(2) -> P^g with modular meaning, whose fibers are birational to M_{0,2g}, the moduli space of 2g-pointed genus zero curves. If g < 4, these modular fibers are even isomorphic to the GIT compactification M^{GIT}_{0,2g}. The families of pointed rational normal curves are recovered as the fibers of the maps that classify extensions of line bundles associated to some effective divisors.

Elliptic Curves in Moduli Space of Stable Bundles

Pure and Applied Mathematics Quarterly, 2011

Let M be the moduli space of rank 2 stable bundles with fixed determinant of degree 1 on a smooth projective curve C of genus g ≥ 2. When C is generic, we show that any elliptic curve on M has degree (respect to anti-canonical divisor −K M) at least 6, and we give a complete classification for elliptic curves of degree 6. Moreover, if g > 4, we show that any elliptic curve passing through the generic point of M has degree at least 12. We also formulate a conjecture for higher rank.

K3 surfaces and equations for Hilbert modular surfaces

Algebra & Number Theory, 2014

We outline a method to compute rational models for the Hilbert modular surfaces Y − (D), which are coarse moduli spaces for principally polarized abelian surfaces with real multiplication by the ring of integers in Q( √ D), via moduli spaces of elliptic K3 surfaces with a Shioda-Inose structure. In particular, we compute equations for all thirty fundamental discriminants D with 1 < D < 100, and analyze rational points and curves on these Hilbert modular surfaces, producing examples of genus-2 curves over Q whose Jacobians have real multiplication over Q.

A Structure Theorem for SUC(2) and the Moduli of Pointed Genus Zero Curves

2009

Let SUC(2) be the moduli space of rank 2 semistable vector bundles with trivial determinant on a smooth complex curve C of genus g> 1,nonhyperellptic if g> 2. In this paper we prove a birational structure theorem for SUC(2) that generalizes that of [Bol07] for genus 2. Notably we give a description of SUC(2) as a fibration over P g, where the fibers are compactifications of the moduli space M0,2g of 2g-pointed genus zero curves. This is done by describing the classifying maps of extensions of the line bundles associated to some effective divisors. In particular, for g = 3 our construction shows that SUC(2) is birational to a fibration in Segre cubics over a P³.

On Some Moduli Spaces of Bundles on K3 Surfaces

Monatshefte Fur Mathematik, 2005

We give infinitely many examples in which the moduli space of rank 2 H-stable sheaves on a K3 surface S endowed by a polarization H of degree 2g – 2, with Chern classes c 1 = H and c 2 = g – 1, is birationally equivalent to the Hilbert scheme S[g – 4] of zero dimensional subschemes of S of length g – 4. We get in this way a partial generalization of results from [5] and [1].

The Picard groups of the moduli spaces of curves

Topology, 1987

the moduli space of smooth h-pointed curves of genus g over C and by &is,h its natural compactification by means of stable curves. It is known that the Picard group of M,,, is a free Abelian group on h + 1 generators when g 2 3. This is due to Harer [4, 51 (cf. the Appendix). Instead of dealing with the Picard group of the moduli space it is usually more convenient, from a technical point of view, to work with the so-called Picard group of the moduli functor (see below for a precise definition), which we shall denote by Pit (JY~,,) if we are restricting to smooth curves and by Pit #&if we are allowing singular stable curves as well. As Mumford observes in [S], Pit (Jg.J has no torsion and contains Pit (MgJ as a subgroup of finite index (a proof of this will be sketched in the Appendix). The purpose of this note is to exhibit explicit bases for Pit (&& and for Pit (d&g,h)r which is also a free Abelian group. This is done in Theorem 2 (53), of which Theorem 1 in $2 is a special case. We shall now say a couple of words about our terminology. A family of h-pointed stable curves of genus g parametrized by S is a proper flat morphism II : V + S together with disjoint sections ol,. .. , a,, having the following properties. Each fiber n-'(s) is a connected curve of genus g having only nodes as singularities and such that each of its smooth rational components contains at least three points belonging to the union of the remaining components and of the sections; moreover, for each i, Go is a smooth point of n-'(s). Following Mumford [7,8], by a line bundle on the moduli functor Jjtg,h we mean the datum of a line bundle L, (often written L,) on S for any family F = (n : %Z + S, (rl,. .. , CT,,) of h-pointed stable curves of genus g, and of an isomorphism L, s cz*(L,) for any Cartesian square of families of h-pointed stable curves; these isomorphisms are moreover required to satisfy an obvious cocycle condition. It is important to notice that we get an equivalent definition if, in the above, we restrict to families of pointed stable curves which are, near any point of the base, universal deformations for the corresponding fiber. We write Pit (Sg,h) to denote the group l Supported in part by grants from the C.N.R. and the Italian Ministry of Public Education.