One tries to understand the isomorphism classes of certain types of geometric objects. More specifically, i would like to know what are interesting problems related to moduli spaces of. This semester, the course aims to introduce techniques for studying intersection theory on moduli spaces. Lectures on logarithmic algebraic geometry by arthur ogus. This site contains activities of the algebraalgebraic geometry group at oakland university. The motivating problems in geometry are classi cation problems. Topics in algebraic geometry tim dokchitser tcc graduate course, spring 2015 contents. The setting is here a smooth ambient algebraic variety x e. Algebraic geometryis a branch of mathematics, classically studying zeros of multivariate polynomials. What are the open big problems in algebraic geometry and vector bundles. Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras over, simplicial commutative rings or. Moduli problems and geometric invariant theory victoria hoskins abstract in this course, we study moduli problems in algebraic geometry and the construction of moduli spaces using geometric invariant theory.
First, one can use it to recover the hodge structures on higher degree cohomology groups from the hodge structure in degree two. In general, moduli theory studies the geometry of families of algebraic objects. Not every functor x comes from an algebra or even a scheme through the yoneda embedding as above. Rationality problems in algebraic geometry pdf rationality problems in algebraic geometry pdf. Now, using a as a coordinate, we can identify now, using a as a coordinate, we can identify vad. The symmetric rank of a monomial can be computed by algebraic tools, and already this elementary case is not trivial. Even in characteristic 0, the cm locus is higher dimensional so the especially stacky locus has high dimension but i dont know how serious the problem is. Topology and geometry of the moduli space of curves pdf. Classical deformation theory in algebraic geometry northeastern. Via stacks the rst part of this thesis is devoted mainly to presenting moduli problems in algebraic geometry using the modern language of stacks. A peculiar feature of algebraic geometry is that there is often a variety or scheme, called the moduli space, whose points correspond to. Categorified hecke algebras, link homology, and hilbert schemes aimpl. We will refer to xas the completion of xat the point. This moduli problem is not representable by a scheme.
This will let us make precise the notion when a certain moduli problem has a fine moduli space. Arthur ogus book, patiently matured and without equivalent today, provides the first systematic study of the subject. If xis gpdvalued, the same formula determines a gpdvalued classical formal moduli problem xhere we take a homotopy ber product of the relevant groupoids. In this course, we study moduli problems in algebraic geometry and the. Stacks and algebraic stacks were invented by the grothendieck school of algebraic geometry in the 1960s. But because polynomials are so ubiquitous in mathematics, algebraic geometry has always stood at the crossroads of many different fields. Algebraic stacks and moduli of vector bundles cimat. The issue of dsemistability does not arise since a nodal curve is automat ically dsemistable. Enumerative algebraic geometry of conics andrew bashelor, amy ksir, and will traves 1. The workshop brought together researchers working on different aspects of moduli theory, to report on progress, discuss open problems, give overview, and in. The focus of the workshop will be the recent progress in derived algebraic geometry, birational geometry and moduli spaces.
In spring 2009, the mathematical sciences research institute will sponsor a program on algebraic geometry, emphasizing crossfertilization between different areas, including classical and complex algebraic geometry, linear series techniques, moduli spaces, enumerative geometry, varieties with group actions, birational geometry, rational curves. Mauro porta, derived formal moduli problems, master thesis 20, pdf. Algebraic geometry is a branch of mathematics that combines techniques of abstract algebra with the language and the problems of geometry. Pdf math, algebra problems, algebra worksheets, algebra. Here you will find information about the mcag conferences, the algebraic database project, our seminars, and the phd program at oakland university. In algebraic geometry, a moduli space is a geometric space usually a scheme or an algebraic stack whose points represent algebrogeometric objects of some. Moduli spaces arise throughout algebraic geometry, differential geometry and algebraic topology. Damien calaque, julien grivaux, formal moduli problems and formal derived stacks arxiv. If xis gpdvalued, the same formula determines a gpdvalued classical formal moduli problem xhere we take a homotopy ber. Contains detailed proofs of the results claimed in my icm address. Idea of space in algebraic geometry a picture book of algebraic geometry 797w spring 04 contents 0. Pages 176 by rita pardini and gian pietro pirola providing an overview of the state of the art on rationality questions in algebraic geometry, this volume gives an update on the most recent developments. A study of formal moduli problems in the setting of commutative and noncommutative derived algebraic geometry. It has a long history, going back more than a thousand years.
These lecture notes consist of an introduction to moduli spaces in algebraic geometry, with a strong emphasis placed on examples related to the. Moduli spaces in topology are often referred to as classifying spaces. The other see 9 and 2 was to develop a more general framework for studying moduli problems. To my surprise, however, some students did read the draft for volume ii and felt it made some contribution to the growing literature of multiple introductions to algebraic geometry. Search for lectures on algebraic geometry i books in the search form now, download or read books for free, just by creating an account to enter our library. So these are more general objects than one would consider in a rst course in algebraic geometry. Algebraic geometry and moduli seminar department of. Philosophically speaking a moduli problem is a classification problem. It has now been four decades since david mumford wrote that algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and.
Setvalued classical formal moduli problem xby the formula xr xr xrm r f g. However i feel that the problems are not very inspiring at the moment at least and lack some depth. In my talk, i will discuss some applications of this fact. Moduli spaces in algebraic geometry mfo, oberwolfach. A set of exercises in order to gain practice with schubert calculus. Lecture notes on quiver representations and moduli.
Idea of space in algebraic geometry a picture book of. More than 1 million books in pdf, epub, mobi, tuebl and audiobook formats. In the last three decades, logarithmic geometry has become a key tool in many areas of arithmetic and algebraic geometry moduli problems, padic hodge theory. Instead, it tries to assemble or, in other words, to. Section4gives the rst example where logarithmic geometry ts well with moduli spaces. A study of closed immersions in spectral algebraic geometry, and the operation of gluing along. Algebraic geometry, moduli spaces, and invariant theory. Algebraic geometry open problems proceedings of the conference held in ravello, may 31 june 5, 1982. It refers to the whole complex of problems associated with the existence of moduli spaces of certain algebraic objects varieties, vector bundles, endomorphisms, etc. I am now using fultons book algebraic curves to learn algebraic geometry from and have just finished chapter 2. All my papers in this field have been published by springerverlag in two volumes, a selected papers on the classification of varieties and moduli spaces, and b selected papers ii, on algebraic geometry including correspondence with grothendieck. The moduli problem frequently receives a broader interpretation see, for example. We rst introduce what a \ moduli problem in algebraic geometry should be formally, that is, through grothendiecks approach using representable functors. Even very simple moduli problems involve the classi cation of geometric objects which admit nontrivial automorphisms, and should therefore be treated as categories rather than as sets examples 1.
A classical moduli problem or prestack is a functor x. It is the latter aspect that interests us in this chapter. One purpose see 11 was to give geometric meaning to higher cohomology classes. Finally, in the last paragraph i have presented a short overview of derived algebraic geometry in. Jacob lurie, moduli problems for ring spectra icm 2010 proceedings contribution pdf. The study materials section contains practice problems on schubert calculus, the hilbert scheme of conics and divisor class calculations for this course.
In x1, we discussed the notion of a moduli problem in classical algebraic geometry. The lectures will be aimed at a wide audience including advanced graduate students and postdocs with a background in algebraic geometry. In 1848 jakob steiner, professor of geometry at the university of berlin, posed the following problem 19. Classify all possible algebraic varieties under some equivalence relation moduli problem. Proceedings of the conference on the moduli space of abelian.
The construction is then generalized to noncommutative geometry. More specifically, i would like to know what are interesting problems related to moduli spaces of vector bundles over projective varietiescurves. The following algebra topics are covered among others. Where is a good source of problems in algebraic geometry that i. The objects of study of algebraic geometry are, roughly, the common zeroes of polynomials in one or several variables algebraic varieties. Therefore, the kind of problems mathematicians try to solve in algebraic geometry are related to much of everything else, mostly. Lecture notes on quiver representations and moduli problems in algebraic geometry. We then explain how to construct group quotients in algebraic geometry via. These worksheets are printable pdf exercises of the highest quality. Rationality problems in algebraic geometry pdf web education.