Nikulin involutions on <Emphasis Type="Italic">K</Emphasis>3 surfaces

We study the maps induced on cohomology by a Nikulin (i.e. a symplectic) involution on a K3 surface. We parametrize the 11-dimensional irreducible components of the moduli space of algebraic K3 surfaces with a Nikulin involution and we give examples of the general K3 surface in various components. We conclude with some remarks on Morrison–Nikulin involutions, these are Nikulin involutions which interchange two copies of E ₈(−1) in the Néron Severi group. The second author is supported by DFG Research Grant SA 1380/1-1.     

On the moduli of surfaces admitting genus 2 fibrations

We investigate the structure of the components of the moduli space of surfaces of general type, which parametrize surfaces admitting nonsmooth genus 2 fibrations of nonalbanese type, over curves of genusg _b≥2.     

The Moduli Space of Curves, Double Hurwitz Numbers, and Faber��s Intersection Number Conjecture

We define the dimension 2g − 1 Faber-Hurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of \mathbbP₁{{\mathbb{P}_1}} with given ramification over ∞ and sufficiently many fixed ramification points elsewhere. Degeneration of the target and judicious localization expresses such classes in terms of localization trees weighted by “top intersections” of tautological classes and genus 0 double Hurwitz numbers. This identity of generating series can be inverted, yielding a “combinatorialization” of top intersections of Y{\Psi} -classes. As genus 0 double Hurwitz numbers with at most 3 parts over ∞ are well understood, we obtain Faber’s Intersection Number Conjecture for up to 3 parts, and an approach to the Conjecture in general (bypassing the Virasoro Conjecture). We also recover other geometric results in a unified manner, including Looijenga’s theorem, the socle theorem for curves with rational tails, and the hyperelliptic locus in terms of κ _g–2.     

Real normalized differentials and Arbarello’s conjecture

Using meromorphic differentials with real periods, we prove Arbarello’s conjecture that any compact complex cycle of dimension g - n in the moduli space M _g of smooth algebraic curves of genus g must intersect the locus of curves having a Weierstrass point of order at most n.     

Surfaces with pg = 2, K2 = 3 and a pencil of curves of genus 2

We classify minimal smooth surfaces of general type with K ² = 3, p _g = 2 which admit a fibration of curves of genus 2.We prove that they form an irreducible set of dimension 22 in their moduli space.      

Pants Decompositions of Random Surfaces

Our goal is to show, in two different contexts, that “random” surfaces have large pants decompositions. First we show that there are hyperbolic surfaces of genus g for which any pants decomposition requires curves of total length at least g ^7/6−ε . Moreover, we prove that this bound holds for most metrics in the moduli space of hyperbolic metrics equipped with the Weil–Petersson volume form. We then consider surfaces obtained by randomly gluing euclidean triangles (with unit side length) together and show that these surfaces have the same property.     

The Prym and Ganning Vector Bundles over the Teichmueller Space

We introduce and study the Prym vector bundle P of holomorphic Prym differentials and the Ganning cohomology bundle G over the Teichmueller space of compact Riemann surfaces of genus g2 and over the Torelli space of genus g2. We construct a basis of holomorphic Prym differentials on a variable compact Riemann surface which depends on the moduli of the compact Riemann surface and on the essential characters. From these bundles we compose an exact sequence of holomorphic vector bundles over the product of the Teichmueller space of genus g and a special domain in the complex manifold C ^2g/Z ^2g.     

Murphy’s law in algebraic geometry: Badly-behaved deformation spaces

We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ? (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over $\mathbb{F}_{p}We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ℤ (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over that lifts to ℤ/p⁷ but not ℤ/p⁸. (Of course the results hold in the holomorphic category as well.) It is usually difficult to compute deformation spaces directly from obstruction theories. We circumvent this by relating them to more tractable deformation spaces via smooth morphisms. The essential starting point is Mn?v’s universality theorem. Mathematics Subject Classification (2000) 14B12, 14C05, 14J10, 14H50, 14B07, 14N20, 14D22, 14B05     

On the rational cohomology of moduli spaces of curves with level structures

We investigate low degree rational cohomology groups of smooth compactifications of moduli spaces of curves with level structures. In particular, we determine H^k([`(S)]<sub>g</sub>, \mathbb Q){H^k\left({\bar S}_{g}, {\mathbb Q}\right)} for g ≥ 2 and k ≤ 3, where [`(S)]_g{{\bar S}_{g}} denotes the moduli space of spin curves of genus g.     

The Deligne–Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3

In the moduli space M \mathcal{M} _g of genus-g Riemann surfaces, consider the locus RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} of Riemann surfaces whose Jacobians have real multiplication by the order O \mathcal{O} in a totally real number field F of degree g. If g = 3, we compute the closure of RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} in the Deligne–Mumford compactification of M \mathcal{M} _g and the closure of the locus of eigenforms over RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} in the Deligne–Mumford compactification of the moduli space of holomorphic 1-forms. For higher genera, we give strong necessary conditions for a stable curve to be in the boundary of RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} . Boundary strata of RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} are parameterized by configurations of elements of the field F satisfying a strong geometry of numbers type restriction.     

Stable degenerations of symmetric squares of curves

The stable (in the sense of the relative minimal model program) degenerations of symmetric squares of smooth curves of genus g>2 are computed. This information is used to prove that the component of the moduli space of stable surfaces parameterizing such surfaces is isomorphic to the moduli space of stable curves of genus g.     

Donaldson theory on non-Kählerian surfaces and class <Emphasis Type="Italic">VII</Emphasis> surfaces with <Emphasis Type="Italic">b</Emphasis><Subscript>2</Subscript>=1

We prove that any class VII surface with b₂=1 has curves. This implies the “Global Spherical Shell conjecture” in the case b₂=1: Any minimal class VII surface withb₂=1 admits a global spherical shell, hence it is isomorphic to one of the surfaces in the known list. By the results in [LYZ], [Te1], which treat the case b₂=0 and give complete proofs of Bogomolov’s theorem, one has a complete classification of all class VII-surfaces with b₂∈{0,1}. The main idea of the proof is to show that a certain moduli space of PU(2)-instantons on a surface X with no curves (if such a surface existed) would contain a closed Riemann surface Y whose general points correspond to non-filtrable holomorphic bundles on X. Then we pass from a family of bundles on X parameterized by Y to a family of bundles on Y parameterized by X, and we use the algebraicity of Y to obtain a contradiction. The proof uses essentially techniques from Donaldson theory: compactness theorems for moduli spaces of PU(2)-instantons and the Kobayashi-Hitchin correspondence on surfaces.     

Triangulations and moduli spaces of Riemann surfaces with group actions

Summary We study that subset of the moduli space of stable genusg,g>1, Riemann surfaces which consists of such stable Riemann surfaces on which a given finite groupF acts. We show first that this subset is compact. It turns out that, for general finite groupsF, the above subset is not connected. We show, however, that for ℤ₂ actions this subsetis connected. Finally, we show that even in the moduli space ofsmooth genusg Riemann surfaces, the subset of those Riemann surfaces on which ℤ₂ actsis connected. In view of deliberations of Klein ([8]), this was somewhat surprising. These results are based on new coordinates for moduli spaces. These coordinates are obtained by certainregular triangulations of Riemann surfaces. These triangulations play an important role also elsewhere, for instance in approximating eigenfunctions of the Laplace operator numerically. This work has been supported by the European Communities Science Plan Project No SCI^*-CT91 (TSTS) “Computational Methods in the Theory of Riemann Surfaces and Algebraic Curves,” by Academy of Finland and by the Swiss National Science Foundation Grant 20-34099.92. We thank M. C. Petrus for providing excellent motivation for this work. This article was processed by the author using the LATEX style filecljourl from Springer-Verlag.     

Curve Shortening Flow in Arbitrary Dimensional Euclidian Space

In this paper, curve shortening flow in Euclidian space R^n（n≥3） is studied, and S. Altschuler＇s results about flow for space curves are generalized. We prove that the curve shortening flow converges to a straight line in infinite time if the initial curve is a ramp. We also prove the planar phenomenon when the curve shortening flow blows up.     

A characterization of bielliptic curves and applications to their moduli spaces

We deal with the covers of degree 4 naturally associated to a bielliptic curve of genus g≥6, giving a proof of the unirationality of the moduli space ? _g ^be of such curves, of the rationality of the Hurwitz scheme ℌ ^be _{4, g} of bielliptic curves of even genus g, whereas, when g is odd, we construct a finite map ℂ^{2 g -2}→? _g ^be and compute its degree. Received: March 25, 2000; in final form: March 10, 2001?Published online: May 29, 2002     

Prym varieties of cyclic coverings

The Prym map of type (g, n, r) associates to every cyclic covering of degree n of a curve of genus g ramified at a reduced divisor of degree r the corresponding Prym variety. We show that the corresponding map of moduli spaces is generically finite in most cases. From this we deduce the dimension of the image of the Prym map.     

Transcendence degree of zero-cycles and the structure of Chow motives

In the paper we introduce a transcendence degree of a zero-cycle on a smooth projective variety X and relate it to the structure of the motive of X. In particular, we show that in order to prove Bloch’s conjecture for a smooth projective complex surface X of general type with p _g = 0 it suffices to prove that one single point of a transcendence degree 2 in X(ℂ), over the minimal subfield of definition k ⊂ ℂ of X, is rationally equivalent to another single point of a transcendence degree zero over k. This can be of particular interest in the context of Bloch’s conjecture for those surfaces which admit a concrete presentation, such as Mumford’s fake surface, see [Mumford D., An algebraic surface with K ample, (K ²) = 9, p _g = q = 0, Amer. J. Math., 1979, 101(1), 233–244].     

Bernstein�CWalsh Type Theorems for Real Analytic Functions in Several Variables

The aim of this paper is to extend the classical maximal convergence theory of Bernstein and Walsh for holomorphic functions in the complex plane to real analytic functions in ℝ ^N . In particular, we investigate the polynomial approximation behavior for functions F:L→ℂ, L={(Re z,Im z):z∈K}, of the structure F=g[`(h)]F=g\overline{h}, where g and h are holomorphic in a neighborhood of a compact set K⊂ℂ <sup>N</sup> . To this end the maximal convergence number ρ(S <sub>c</sub> ,f) for continuous functions f defined on a compact set S <sub>c</sub> ⊂ℂ <sup>N</sup> is connected to a maximal convergence number ρ(S <sub>r</sub> ,F) for continuous functions F defined on a compact set S <sub>r</sub> ⊂ℝ <sup>N</sup> . We prove that ρ(L,F)=min {ρ(K,h)),ρ(K,g)} for functions F=g[`(h)]F=g\overline{h} if K is either a closed Euclidean ball or a closed polydisc. Furthermore, we show that min {ρ(K,h)),ρ(K,g)}≤ρ(L,F) if K is regular in the sense of pluripotential theory and equality does not hold in general. Our results are based on the theory of the pluricomplex Green’s function with pole at infinity and Lundin’s formula for Siciak’s extremal function Φ. A properly chosen transformation of Joukowski type plays an important role.