The topics for this course vary each semester. This semester, the course aims to introduce techniques for studying intersection theory on moduli spaces.
Intersection theory on moduli spaces. Image by Dr. Instructor s Dr. Some Description Instructor s Prof.
Communications in Mathematical Physics
Need help getting started? Don't show me this again Welcome! The content varies year to year, according to the interests of the instructor and the students. It is frequently the case that interesting geometric objects come equipped with lots of natural automorphisms.
Modern Moduli Theory
However, this approach is not ideal, as such spaces are not guaranteed to exist, they are frequently singular when they do exist, and miss details about some non-trivial families of objects they classify. A more sophisticated approach is to enrich the classification by remembering the isomorphisms. More precisely, on any base B one can consider the category of families on B with only isomorphisms between families taken as morphisms. One then considers the fibred category which assigns to any space B the groupoid of families over B. In general, they cannot be represented by schemes or even algebraic spaces , but in many cases, they have a natural structure of an algebraic stack.
Algebraic stacks and their use to analyze moduli problems appeared in Deligne-Mumford as a tool to prove the irreducibility of the coarse moduli space of curves of a given genus. The language of algebraic stacks essentially provides a systematic way to view the fibred category that constitutes the moduli problem as a "space", and the moduli stack of many moduli problems is better-behaved such as smooth than the corresponding coarse moduli space.
A curve is stable if it has only a finite group of automorphisms. Both moduli stacks carry universal families of curves. One can also define coarse moduli spaces representing isomorphism classes of smooth or stable curves. These coarse moduli spaces were actually studied before the notion of moduli stack was invented.
In fact, the idea of a moduli stack was invented by Deligne and Mumford in an attempt to prove the projectivity of the coarse moduli spaces. In recent years, it has become apparent that the stack of curves is actually the more fundamental object. In lower genus, one must account for the presence of smooth families of automorphisms, by subtracting their number.
There is exactly one complex curve of genus zero, the Riemann sphere, and its group of isomorphisms is PGL 2. Likewise, in genus 1, there is a one-dimensional space of curves, but every such curve has a one-dimensional group of automorphisms. Eventually, in genus zero, the coarse moduli space has dimension zero, and in genus one, it has dimension one. One can also enrich the problem by considering the moduli stack of genus g nodal curves with n marked points. Such marked curves are said to be stable if the subgroup of curve automorphisms which fix the marked points is finite.
This is the stack of elliptic curves , and is the natural home of the much studied modular forms , which are meromorphic sections of bundles on this stack. In higher dimensions, moduli of algebraic varieties are more difficult to construct and study. For instance, the higher-dimensional analogue of the moduli space of elliptic curves discussed above is the moduli space of abelian varieties, such as the Siegel modular variety.
This is the problem underlying Siegel modular form theory. See also Shimura variety. Another important moduli problem is to understand the geometry of various substacks of the moduli stack Vect n X of rank n vector bundles on a fixed algebraic variety X.
This stack has been most studied when X is one-dimensional, and especially when n equals one. In this case, the coarse moduli space is the Picard scheme , which like the moduli space of curves, was studied before stacks were invented. When the bundles have rank 1 and degree zero, the study of coarse moduli space is the study of the Jacobian variety. In applications to physics , the number of moduli of vector bundles and the closely related problem of the number of moduli of principal G-bundles has been found to be significant in gauge theory.
The talks, in particular, describe the general method of constructing moduli spaces by first rigidifying the moduli problem under consideration. More precisely, the existence of non-trivial automorphisms of the objects being classified makes it impossible to have a fine moduli space.
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However, it is often possible to consider a modified moduli problem of classifying the original objects together with additional data, chosen in such a way that the identity is the only automorphism respecting also the additional data. With a suitable choice of the rigidifying data, the modified moduli problem will have a fine moduli space T , often described as a subscheme of a suitable Hilbert scheme or Quot scheme.
The rigidifying data is moreover chosen so that it corresponds to a principal bundle with an algebraic structure group G. The last problem, in general, does not admit a solution; however, it is addressed by the groundbreaking geometric invariant theory GIT , developed by David Mumford in , which shows that under suitable conditions the quotient indeed exists. Consequently, the moduli space of smooth curves and linear systems satisfying certain criteria may be embedded in the Hilbert scheme of a sufficiently high-dimensional projective space.
Category:Moduli theory - Wikipedia
This locus H in the Hilbert scheme has an action of PGL n which mixes the elements of the linear system; consequently, the moduli space of smooth curves is then recovered as the quotient of H by the projective general linear group. Another general approach is primarily associated with Michael Artin. Here the idea is to start with an object of the kind to be classified and study its deformation theory.
This means first constructing infinitesimal deformations, then appealing to prorepresentability theorems to put these together into an object over a formal base. Next, an appeal to Grothendieck's formal existence theorem provides an object of the desired kind over a base which is a complete local ring. This object can be approximated via Artin's approximation theorem by an object defined over a finitely generated ring.
The spectrum of this latter ring can then be viewed as giving a kind of coordinate chart on the desired moduli space. By gluing together enough of these charts, we can cover the space, but the map from our union of spectra to the moduli space will, in general, be many to one. We, therefore, define an equivalence relation on the former; essentially, two points are equivalent if the objects over each are isomorphic. This gives a scheme and an equivalence relation, which is enough to define an algebraic space actually an algebraic stack if we are being careful if not always a scheme.
The term moduli space is sometimes used in physics to refer specifically to the moduli space of vacuum expectation values of a set of scalar fields , or to the moduli space of possible string backgrounds. Moduli spaces also appear in physics in cohomological field theory , where one can use Feynman path integrals to compute the intersection numbers of various algebraic moduli spaces.
From Wikipedia, the free encyclopedia. Further information: Moduli of algebraic curves.