There are various methods of constructing quotients in algebraic geometry. The lecture will concentrate on Mumford’s Geometric Invariant Theory (GIT).
References
P. E. Newstead, Introduction to Moduli Problems and Orbit Spaces, Tata Institute Lecture Notes on Mathematics, Vol 51, 1978.
I. Dolgachev, Introduction to Geometric Invariant Theory, Lecture Note series 25, Seoul National University, Research Institute of Mathematics Global Analysis Research Centre, Seoul, 1994.
D. Mumford, J. Fogarty and F. Kirwan, Geometric Invariant Theory, 3rd. edition, Springer-Verlag, Berlin, 1994.
The basic classification of vector bundles on algebraic curves was carried out 40 years ago. The lecture will describe this classification in broad terms (more details in Lecture 3) and also survey what is now known about the moduli spaces used to classify bundles (more details in Lectures 3 and 4).
References
P. E. Newstead, Topological properties of some spaces of stable bundles, Topology 6 (1967), 241-262
P. E. Newstead, Characteristic classes of stable bundles of rank 2 over an algebraic curve, Trans. Amer. Math. Soc. 169 (1972), 337-345
M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. London A308 (1982), 523-615.
F. Kirwan, The cohomology ring of moduli spaces of bundles over Riemann surfaces, J. Amer. Math. Soc. 5 (1992), 853-906
M. Thaddeus, Conformal field theory and the cohomology of the moduli space of stable bundles, J. Diff. Geom. 35 (1992), 131-149
D. Zagier, On the cohomology of moduli spaces of rank two vector bundles over curves, Progr. Math. 129 (1995), 533-563
V. Yu. Baranovskii, Cohomology ring of the moduli space of stable vector bundles with odd determinant, Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), 204-210
A. D. King and P. E. Newstead, On the cohomology ring of the moduli space of rank 2 vector bundles on a curve, Topology 37 (1998), 407-418
B. Siebert and G. Tian, Recursive relations for the cohomology ring of moduli spaces of stable bundles, Turkish J. Math. 19 (1996), 131-144
R. Herrera and S. Salamon, Intersection numbers on moduli spaces and symmetries of a Verlinde formula, Comm. Math. Phys. 188 (1997), 521-534
Much is now known about the topology of the moduli spaces, especially their cohomology. This lecture will describe some of the methods used to obtain this information. This aspect of moduli spaces has been of much interest to theoretical physicists in connection with Seiberg-Witten invariants and similar computations.
References
P. E. Newstead, Topological properties of some spaces of stable bundles, Topology 6 (1967), 241-262.
P. E. Newstead, Characteristic classes of stable bundles of rank 2 over an algebraic curve, Trans. Amer. Math. Soc. 169 (1972), 337-345.
M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. London A308 (1982), 523-615.
F. Kirwan, The cohomology ring of moduli spaces of bundles over Riemann surfaces, J. Amer. Math. Soc. 5 (1992), 853-906.
M. Thaddeus, Conformal field theory and the cohomology of the moduli space of stable bundles, J. Diff. Geom. 35 (1992), 131-149.
D. Zagier, On the cohomology of moduli spaces of rank two vector bundles over curves, Progr. Math. 129 (1995), 533-563.
V. Yu. Baranovskii, Cohomology ring of the moduli space of stable vector bundles with odd determinant, Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), 204-210.
A. D. King and P. E. Newstead, On the cohomology ring of the moduli space of rank 2 vector bundles on a curve, Topology 37 (1998), 407-418.
B. Siebert and G. Tian, Recursive relations for the cohomology ring of moduli spaces of stable bundles, Turkish J. Math. 19 (1996), 131-144.
R. Herrera and S. Salamon, Intersection numbers on moduli spaces and symmetries of a Verlinde formula, Comm. Math. Phys. 188 (1997), 521-534.
The geometry of the moduli spaces has also been studied, though perhaps less thoroughly than the topology. This lecture will describe some of these geometrical aspects, especially the Segre stratification and Brill-Noether theory.
References
A. King and A. Schofield, Rationality of moduli of vector bundles on curves, Indag. Math. (N.S.) 10 (1999), 519-535.
U. N. Bhosle, Moduli of orthogonal and spin bundles on hyperelliptic curves, Comp. Math. 51 (1984), 15-40.
M. Teixidor i Bigas, Brill-Noether theory for stable vector bundles, Duke Math. J. 62 (1991), 385-400.
L. Brambila-Paz, I. Grzegorczyk and P. E. Newstead, Geography of Brill-Noether loci for small slopes, J. Alg. Geom. 6 (1997), 645-669.
V. Mercat, Le probleme de Brill-Noether pour les fibres stables de petite pente, J. Reine Angew. Math. 506 (1999), 1-14.
L. Brambila-Paz, V. Mercat. P. E. Newstead and F. Ongay, Nonemptiness of Brill-Noether loci, Internat. J. Math. 11 (2000), 737-760.
H. Lange and M. S. Narasimhan, Maximal subbundles of rank 2 vector bundles on curves, Math. Ann. 266 (1983), 55-72.
L. Brambila-Paz and H. Lange, A stratification of the moduli space of vector bundles on curves, J. Reine Angew. Math. 499 (1998), 173-187.
B. Russo and M. Teixidor i Bigas, On a conjecture of Lange, J. Alg. Geom. 8 (1999), 483-496.
