Pullback Moduli Spaces

Geometric invariant theory can be used to construct moduli spaces associated to representations of finite dimensional algebras. One difficulty which occurs in various natural cases is that nonisomorphic modules are sent to the same point in the moduli spaces which arise. In this article, we study how this collapsing phenomenon can sometimes be reduced by considering pullbacks of modules for an auxiliary algebra. One application is a geometric proof that the twisting action of an algebra automorphism induces an algebraic isomorphism between moduli spaces.     

Ramanujan's Singular Moduli

In his first notebook, in scattered places, Ramanujan recorded without proofs the values of over 100 class invariants and over 30 singular moduli. This paper is devoted to establishing all of Ramanujan's representations for singular moduli. For those of odd index, new algorithms needed to be developed.     

Moduli of Stable Surfaces

This article is a survey of basic facts about the moduli spaces of stable surfaces. These spaces are projective compactifications of the moduli spaces of minimal surfaces of general type. They share few of the nice features of the moduli spaces of stable curves. Some of the main pathologies are presented with some historical remarks and some more recent results on the boundary of these spaces.Received: February, 2004     

Moduli uniseriali non standard

Vengono esposti i risutati ottenuti per i moduli uniseriali non standard su domini di valutazione, a partire dalla prima dimostrazione della loro esistenza, data da Shelah nel 1984: le altre prove di esistenza, la loro struttura interna, l'organizzazione in una certa gerarchia ed alcune applicazioni, tra cui la soluzione di un problema posto da Kaplansky ed una generalizzazione della classica equivalenza di Matlis. The results obtained for non standard uniserial modules over valuation domains after the first proof of their existence given by Shelah in 1984 are exposed: the other existence proofs, their internal structure, the organization of their class in a suitable hierarchy, and some applications, as the solution of a problem posed by Kaplansky and a generalization of the classical equivalence.

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(Conferenza tenuta il 20 maggio 1990)     

Moduli of suspension spectra

For a -connected spectrum , we study the moduli space of suspension spectra which come equipped with a weak equivalence to . We construct a spectral sequence converging to the homotopy of the moduli space in positive degrees. In the metastable range, we get a complete homotopical classification of the path components of the moduli space. Our main tool is Goodwillie's calculus of homotopy functors.

     

Wilson Quotients for Composite Moduli

An analogue for composite moduli of the Wilson quotient is studied. Various congruences are derived, and the question of when these quotients are divisible by is investigated; such an will be called a ``Wilson number". It is shown that numbers in certain infinite classes cannot be Wilson numbers. Eight new Wilson numbers up to 500 million were found.

     

Tropicalization of Classical Moduli Spaces

The image of the complement of a hyperplane arrangement under a monomial map can be tropicalized combinatorially using matroid theory. We apply this to classical moduli spaces that are associated with complex reflection arrangements. Starting from modular curves, we visit the Segre cubic, the Igusa quartic, and moduli of marked del Pezzo surfaces of degrees 2 and 3. Our primary example is the Burkhardt quartic, whose tropicalization is a 3-dimensional fan in 39-dimensional space. This effectuates a synthesis of concrete and abstract approaches to tropical moduli of genus 2 curves.     

Moduli spaces of quadratic differentials

The cotangent bundle ofJ (g, n) is a union of complex analytic subvarieties, V(π), the level sets of the function “singularity pattern” of quadratic differentials. Each V(π) is endowed with a natural affine complex structure and volume element. The latter contracts to a real analytic volume element, Μ_π, on the unit hypersurface, V₁(π), for the Teichmüller metric. Μ_π is invariant under the pure mapping class group, γ(g, n), and a certain class of functions is proved to be L^p(Μ_π), 0 <p < 1, over the moduli space V₁(π)/γ (g, n). In particular, Μ_π(V₁(π)/γ(g, n)) < ∞, a statement which generalizes a theorem by H. Masur. Research supported by NSF-MCS-8219148 and NSF-DMS-8521620.     

Moduli of continuity and rearrangements

It is proved that the modulus of continuity of a function in L_p(0, 1) majorizes the modulus of continuity of its decreasing rearrangement.     

Moduli of twisted orbifold sheaves

We study stacks of slope-semistable twisted sheaves on orbisurfaces with projective coarse spaces and prove that in certain cases they have many of the asymptotic properties enjoyed by the moduli of slope-semistable sheaves on smooth projective surfaces.     

Theta Characteristics and Their Moduli

We discuss topics related to the geometry of theta characteristics on algebraic curves. They include the birational classification of the moduli space S _g of spin curves of genus g, interpretation of theta characteristics as quadrics in a vector space over the field with two elements as well as the connection with modular forms and superstring scattering amplitudes. Special attention is paid to the historical development of the subject.     

Moduli dimensions of lattice polygons

Journal of Algebraic Combinatorics - Given a lattice polygon P with g interior lattice points, we can associate to $$P$$ two moduli spaces: the moduli space of algebraic curves that are...     

Moduli of polarized Hodge structures

Around 1970 Griffiths introduced the moduli of polarized Hodge structures/ the period domain D and described a dream to enlarge D to a moduli space of degenerating polarized Hodge structures. Since in general D is not a Hermitian symmetric domain, he asked for the existence of a certain automorphic cohomology theory for D, generalizing the usual notion of automorphic forms on symmetric Hermitian domains. Since then there have been many efforts in the first part of Griffith's dream but the second part still lives in darkness. The objective of the present text is two-folded. First, we give an exposition of the subject. Second, we give another formulation of the Griffiths problem, based on the classical Weierstrass uniformization theorem.     

Moduli of Framed Parabolic Sheaves

This paper gives a construction of the moduli space of framed parabolic sheaves on a Riemann surface. This space serves as a universal, master, space for the well known moduli space of parabolic bundles, as well as moduli spaces of vector bundles, which can all be obtained from this space by torus quotients. The construction is given for the structure group SL(N, C), and indeed is adapted to this case. At the end of the paper, an approach is suggested for dealing with the case of arbitrary reductive groups, involving the associated loop group.     

Moduli of Polarised Abelian Surfaces

The moduli variety parametrising abelian surfaces over ? with a polarisation of type (1, p) may be compactified by toroidal methods. We show that if p is a prime and p ≥ 173 then the compactified moduli variety is of general type.