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This book focusses on a large class of objects in moduli theory and provides different perspectives from which compactifications of moduli spaces may be investigated.

Three contributions give an insight on particular aspects of moduli problems. In the first of them, various ways to construct and compactify moduli spaces are presented. In the second, some questions on the boundary of moduli spaces of surfaces are addressed. Finally, the theory of stable quotients is explained, which yields meaningful compactifications of moduli spaces of maps.

Both advanced graduate students and researchers in algebraic geometry will find this book a valuable read.



Chapter 1. Perspectives on the Construction and Compactification of Moduli Spaces

A central theme in algebraic geometry is the construction of compact moduli spaces with geometric meaning. The two early successes of the moduli theory – the construction and compactification of the moduli spaces of curves \( \bar M_g \) and principally polarized abelian varieties (ppavs) \( \bar A_g \) – are models that we try to emulate. While very few other examples are so well understood, the tools developed to study other moduli spaces have led to new developments and unexpected directions in algebraic geometry. The purpose of these notes is to review three standard approaches to constructing and compactifying moduli spaces: GIT, Hodge theory, and MMP, and to discuss various connections between them.
Radu Laza

Chapter 2. Compact Moduli Spaces of Surfaces and Exceptional Vector Bundles

As shown by Kollár and Shepherd-Barron [17] the moduli space of surfaces of general type has a natural compactification, which is analogous to the Deligne–Mumford compactification of the moduli space of curves [5]. However, very little is known about this moduli space or its compactification in general (for example it can have many irreducible components [3] and be highly singular [29]). A key question is to enumerate the boundary divisors in cases where the moduli space is well behaved. The most basic boundary divisors are those given by degenerations of the smooth surface to a surface with a cyclic quotient singularity of a special type, first studied by J. Wahl [30]. We describe a construction which relates these boundary divisors to the classification of stable vector bundles on the smooth surface in the case \( H^{2,\,0} = H^1 = 0 \). In particular, we connect with the theory of exceptional collections of vector bundles used in the study of the derived category of coherent sheaves.
Paul Hacking

Chapter 3. Notes on the Moduli Space of Stable Quotients

In this chapter, we consider compactifications of the space of maps from curves to Grassmannians and to related geometries. We focus on sheaf theoretic compactifications via Quot scheme-type constructions. An underlying theme is the geometric description of such compactifications, as well as the comparison of the intersection theoretic invariants arising from them.
Dragos Oprea
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