Classifying spaces for polarized mixed Hodge structures and for Brieskorn lattices

Classifying spaces and moduli spaces are constructed for two invariants of isolated hypersurface singularities, for the polarized mixed Hodge structure on the middle cohomology of the Milnor fibre, and for the Brieskorn lattice as a subspace of the Gauß–Manin connection. The relations between them, period mappings for -constant families of singularities, and Torelli theorems are discussed.     

Extracting invariants of isolated hypersurface singularities from their moduli algebras

We use classical invariant theory to construct invariants of complex graded Gorenstein algebras of finite vector space dimension. As a consequence, we obtain a way of extracting certain numerical invariants of quasi-homogeneous isolated hypersurface singularities from their moduli algebras, which extends an earlier result due to the first author. Furthermore, we conjecture that the invariants so constructed solve the biholomorphic equivalence problem in the homogeneous case. The conjecture is easily verified for binary quartics and ternary cubics. We show that it also holds for binary quintics and sextics. In the latter cases the proofs are much more involved. In particular, we provide a complete list of canonical forms of binary sextics, which is a result of independent interest.     

A remark on the liftable derivation of moduli algebras of isolated hypersurface singularities

An example is given to show that not every derivation in the nilradical of the Lie algebra of derivations of moduli algebras can be liftable and the dimension of the nilradical of the Lie algebra of derivations of moduli algebras is not a topological invariant for an isolated hypersurface singularity.

     

Versal Deformations in Spaces of Polynomials of Fixed Weight

This work was largely inspired by a paper of Shustin, in which he proves that for a plane curve of given degree n whose singularities are not too complicated the singularities are versally unfolded by embedding the curve in the space of all curves of degree n; however, our methods are very different. The main result gives fairly explicit lower bounds on the sum of the Tjurina numbers at the singularities of a deformation of a weighted-homogeneous hypersurface, when the deformation is the fibre over an unstable point of an appropriate unfolding. The result is sufficiently flexible to cover a variety of applications, some of which we describe. In particular, we will deduce a generalisation of Shustin's result. Properties of discriminant matrices of unfoldings of weighted-homogeneous functions are crucial to the arguments; the parts of the theory needed are described.     

Focal varieties of curves of genus 6 and 8

In this paper we give a simple Torelli type theorem for curves of genus 6 and 8 by showing that these curves can be reconstructed from their Brill Noether varieties. Among other results, it is shown that the focal variety of a general, canonical and nonhyperelliptic curve of genus 6, is a hypersurface.     

The Aluffi algebra of hypersurfaces with isolated singularities

The Alu? algebra is an algebraic definition of a characteristic cycle of a hypersurface in intersection theory. In this paper, we study the Alu? algebra of quasi-homogeneous and locally Eulerian hypersurfaces with only isolated singularities. We prove that the Jacobian ideal of an a?ne hypersurface with isolated singularities is of linear type if and only if it is locally Eulerian. We show that the gradient ideal of a projective hypersurface with only isolated singularities is of linear type if and only if the a?ne curve in each a?ne chart associated to singular points is locally Eulerian. We show that the gradient ideal of Nodal and Cuspidal projective plane curves are of linear type.     

Application of classical invariant theory to biholomorphic classification of plane curve singularities,and associated binary forms

We use classical invariant theory to solve the biholomorphic equivalence problem for two families of plane curve singularities previously considered in the literature. Our calculations motivate an intriguing conjecture that proposes a method for extracting a complete set of invariants of homogeneous plane curve singularities from their moduli algebras.     

On the tropical Torelli map

We construct the moduli spaces of tropical curves and tropical principally polarized abelian varieties, working in the category of (what we call) stacky fans. We define the tropical Torelli map between these two moduli spaces and we study the fibers (tropical Torelli theorem) and the image of this map (tropical Schottky problem). Finally we determine the image of the planar tropical curves via the tropical Torelli map and we use it to give a positive answer to a question raised by Namikawa on the compactified classical Torelli map.     

Torelli Theorem for the Moduli Spaces of Rank 2 Quadratic Pairs

Let X be a smooth projective complex curve. We prove that a Torelli type theorem holds, under certain conditions, for the moduli space of α-polystable quadratic pairs on X of rank 2.     

The Kodaira dimension of the moduli of K3 surfaces

The global Torelli theorem for projective K3 surfaces was first proved by Piatetskii-Shapiro and Shafarevich 35 years ago, opening the way to treating moduli problems for K3 surfaces. The moduli space of polarised K3 surfaces of degree 2d is a quasi-projective variety of dimension 19. For general d very little has been known hitherto about the Kodaira dimension of these varieties. In this paper we present an almost complete solution to this problem. Our main result says that this moduli space is of general type for d>61 and for d=46, 50, 54, 57, 58, 60.     

On the syzygies and Hodge theory of nodal hypersurfaces

We give lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining an even dimensional nodal hypersurface. This implies the validity of formulas due to M. Saito, L. Wotzlaw and the author for the graded pieces with respect to the Hodge filtration of the top cohomology of the hypersurface complement in many new cases. A classical result by Severi on the position of the singularities of a nodal surface in \(\mathbb {P}^3\) is improved and applications to deformation theory of nodal surfaces are given.     

Interaction of Conormal Waves with Different Singularities for Semi-linear Equations

We consider a solution of the semi-linear partial differential equations in higher space dimensions. We show that if there exist two characteristic hypersurface bearing different weak singularities intersect transversally, and another one characteristic hypersurface issues from above intersection, then the solution would be conormal with respect to the union of these surfaces, and satisfy the so-called “sum law”.     

Universal Deformation Formulas

We give a conceptual explanation of universal deformation formulas for unital associative algebras and prove some results on the structure of their moduli spaces. We then generalize universal deformation formulas to other types of algebras and their diagrams.     

Moduli algebras of some non-semiquasihomogeneous singularities

Under some additional restrictions we find dimensions and bases of moduli algebras of isolated singularities of polynomials in n variables that are sums of n monomials of equal weighted degrees and one monomial of lower degree.     

Moduli of products of curves

Some technical results on the deformations of varieties of general type and on permanence of semi-log-canonical singularities are proved. These results are applied to show that the connected component of the moduli space of stable surfaces containing the moduli point of a product of stable curves is the product of the moduli spaces of the curves, assuming the curves have different genera. An application of this result shows that even after compactifying the moduli space and fixing numerical invariants, the moduli spaces are still very disconnected.Received: 20 February 2004     

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     

Differential Algebras with Dense Singularities on Manifolds

Recently the space-time foam differential algebras of generalized functions with dense singularities were introduced, motivated by the so called space-time foam structures in General Relativity with dense singularities, and by Quantum Gravity. A variety of applications of these algebras has been presented, among them, a global Cauchy-Kovalevskaia theorem, de Rham cohomology in abstract differential geometry, and so on. So far the space-time foam algebras have only been constructed on Euclidean spaces. In this paper, owing to their relevance in General Relativity among others, the construction of these algebras is extended to arbitrary finite dimensional smooth manifolds. Since these algebras contain the Schwartz distributions, the extension of their construction to manifolds also solves the long outstanding problem of defining distributions on manifolds, and doing so in ways compatible with nonlinear operations. Earlier, similar attempts were made in the literature with respect to the extension of the Colombeau algebras to manifolds, algebras which also contain the distributions. These attempts have encountered significant technical difficulties, owing to the growth condition type limitations the elements of Colombeau algebras have to satisfy near singularities. Since in this paper no any type of such or other growth conditions are required in the construction of space-time foam algebras, their extension to manifolds proceeds in a surprisingly easy and natural way. It is also shown that the space-time foam algebras form a fine and flabby sheaf, properties which are important in securing a considerably large class of singularities which generalized functions can handle.     

Variation homological diagrams,¶and corner singularities

We describe the general homological framework (the variation arrays and variation homological diagrams) in which can be studied hypersurface isolated singularities as well as boundary singularities and corner singularities from the point of view of duality. We then show that any corner singularity is extension, in a sense which is defined, of the corner singularities of less dimension on which it is built. This framework is also used to rewrite Thom–Sebastiani type properties for isolated singularities and to establish them for boundary singularities. Received: 27 June 2000 / Revised version: 18 October 2000     

Tilting Theoretical Approach to Moduli Spaces Over Preprojective Algebras

We apply tilting theory over preprojective algebras Λ to the study of moduli spaces of Λ-modules. We define the categories of semistable modules and give equivalences, so-called reflection functors, between them by using tilting modules over Λ. Moreover we prove that the equivalence induces an isomorphism of K-schemes between moduli spaces. In particular, we study the case when the moduli spaces are related to Kleinian singularities, and generalize some results of Crawley-Boevey (Am J Math 122:1027–1037, 2000).