# Seminar on Moduli spaces

We will be running a seminar on moduli spaces of stable things (quiver representations, sheaves) during the autumn 2013 semester.

The meetings will be Wednesdays, 3pm, in JCMB 5215 (note change of room).

## Proposed Outline

Classically, moduli spaces of stable sheaves were constructed as group quotients using GIT (Geometric Invariant Theory). However, in order to understand them explicitly as projective varieties, this abstract construction isn't good enough: one also needs to construct explicit sections of line bundles. There is a well-known way to construct such sections that goes back to work of Faltings (as part of his work to give a GIT-free construction of moduli spaces of stable bundles on curves); this construction now appears under a variety of names in the literature (effective GIT, generalized $$\theta$$-functions, homologically orthogonal bundles, strange duality, generic vanishing, interpolation).
There is a completely parallel problem in the theory of quiver representations, which has essentially been solved in the work of Schofield, van den Bergh, Derksen, Weyman. This was applied by Álvarez-Cónsul, King to solve several open problem about moduli space of stable sheaves, based on the observation that any moduli space of stable sheaves can be embedded in a moduli space of stable representations of the so-called Kronecker quiver. This connection may well have many unexplored applications.

See (New02) for Newstead's notes with a nice short introduction to moduli of vector bundles on algebraic curves. (HL) is the classical book studying stability and moduli of sheaves with emphasis on sheaves on a surface. Chapter 10 of Huybrechts online lecture notes on K3 surfaces (H) deals with moduli of sheaves on K3's.

List of talks:

1. Moduli spaces of stable quiver representations: construction, definition of stability, examples. (Rei08), section 3-4 (but with more examples added).
2. Slope-stability con curves: definition, existence of HN-filtrations, examples (g=0, g=1, rk=1, ...), two families containing all semistable bundles of given dimension vector (for rank 2 see (Hei07); for quot scheme any of the standard references); S-equivalence.
3. Classical construction of Gieseker-moduli spaces via GIT
4. Functorial construction as in (A-CK07)
5. Overview talk: sections of line bundles on moduli spaces
6. Faltings' GIT-free construction of moduli space of stable bundles, see (Hei07)
7. Theory of semi-invariants for quiver representations...
8. ...continued
9. Applications to algebraic geometry (A-CK09)...
10. ...continued

Tentative list of speakers: 1. David, 2. Becca, 3. Johan, 4. Ciaran, 5. Arend, 6. Evgeny, 7/8. Natalia/Milena, 9./10 Joe/Hendrik

### References

• (A-CK09) Álvarez-Cónsul, Luis; King, Alastair: Moduli of sheaves from moduli of Kronecker modules. Moduli spaces and vector bundles, 212–228, London Math. Soc. Lecture Note Ser., 359, Cambridge Univ. Press, Cambridge, 2009. 09
• (A-CK07) Álvarez-Cónsul, Luis; King, Alastair: A functorial construction of moduli of sheaves. Invent. Math. 168 (2007), no. 3, 613–666.
• (DW00) Derksen, Harm; Weyman, Jerzy: Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients. J. Amer. Math. Soc. 13 (2000), no. 3, 467–479 (electronic).
• (Hei07) Hein, Georg: Faltings' construction of the moduli space of vector bundles on a smooth projective curve. http://www.uni-due.de/~hm0019/math/pdf/workshop.pdf
• (H) Huybrechts, D.: Lectures on K3 surfaces. http://www.math.uni-bonn.de/people/huybrech/K3.html
• (HL) Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves.
• (King94) King, A. D.: Moduli of representations of finite-dimensional algebras. Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 515–530.
• (Muk03) Mukai, Shigeru: An introduction to invariants and moduli. Translated from the 1998 and 2000 Japanese editions by W. M. Oxbury. Cambridge Studies in Advanced Mathematics, 81. Cambridge University Press, Cambridge, 2003. xx+503 pp. ISBN: 0-521-80906-1.
• (New78) Newstead, P. E.: Introduction to moduli problems and orbit spaces. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 51. Tata Institute of Fundamental Research, Bombay; by the Narosa Publishing House, New Delhi, 1978. vi+183 pp. ISBN: 0-387-08851-2.
• (New02) Newstead, P.E.: Vector Bundles on Algebraic Curves, http://www.mimuw.edu.pl/~jarekw/postscript/Lukecin-Newstead.ps
• (Rei08) Reineke, Markus: Moduli of representations of quivers. Trends in representation theory of algebras and related topics, 589–637, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2008.
• (SvdB01) Schofield, Aidan; van den Bergh, Michel Semi-invariants of quivers for arbitrary dimension vectors. Indag. Math. (N.S.) 12 (2001), no. 1, 125–138

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