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Handbook of Moduli, Volume I

Gavril Farkas and Ian Morrison, editors
International Press
Publication Date: 
Number of Pages: 
Advanced Lectures in Mathematics 24
[Reviewed by
Fernando Q. Gouvêa
, on

Algebraic geometers have been thinking about moduli spaces for a very long time, and the topic has a central role in that discipline and in various other branches of mathematics. Basically, a moduli space is some sort of geometric object whose points classify (equivalence classes of) other geometric objects in a way that makes sense of “continuously varying” the objects being classified.

David Ben-Zvi has a very nice article about moduli spaces in the Princeton Companion to Mathematics. Here, let’s only consider two easy examples. Suppose, first, that the objects we want to classify are lines on the plane. We remember that any such line is given by an equation of the form \(ax+by=c\), and so we can represent the line by the triple \((a,b,c)\). In order to have an honest line, we need \(a\) and \(b\) not to be both zero. And we need to identify two triples that differ by scaling, since \((ka,kb,kc)\) defines the same line as \((a,b,c)\) for any \(k\neq 0\).

The set of all triples \((a,b,c)\neq (0,0,0)\), taken up to scaling, is the real projective plane \(\mathbb{P}^2(\mathbb{R})\). The only point excluded by our requirement that \(a\) and \(b\) can’t both be zero is \((0,0,1)\), so the “space of lines in \(\mathbb{R}^2\)” is \(\mathbb{P}^2(\mathbb{R})\) minus one point. It is, of course, very tempting to add that extra line somehow, thereby compactifying the space of all lines. And, since \(\mathbb{P}^2(\mathbb{R})\) has a natural topology, we can talk about a continuous (or algebraic) family of lines: it is just a continuous (or algebraic) curve in \(\mathbb{P}^2(\mathbb{R})\).

In most cases, we don’t actually want to consider all objects in a class, but rather want to consider them up to some kind of equivalence. Here is a famous example. Suppose we start by considering four points in the extended complex plane \(\mathbb{C}\cup\infty\). Classifying all such sets of four points is easy (and dumb): the resulting space is just the Cartesian product of four copies of the extended plane. What is interesting, on the other hand is to bring in the standard Möbius transformations \[z\mapsto\frac{az+b}{cz+d}\] and to say two quadruples are equivalent if there exists a Möbius transformation taking one to the other. It turns out that any three points can always be transformed to the points \((0,1,\infty)\), so here’s what we do: given any quadruple \((z_1,z_2,z_3,z_4)\), find a Möbius transformation that sends the first three to \(0\) ,\(1\), and \(\infty\). The quadruple will become \((0,1,\infty, w)\), and the number \(w\) turns out to classify all quadruples up to this equivalence. It can be any complex number except for \(0\), \(1\), and \(\infty\), so once again we have a moduli space that is a “punctured” compact space.

Those examples are easy, but the things people actually think about are very, very hard. So hard, in fact, that the subject runs the risk of splitting up into small groups, each of which work on a particular kind of moduli problem. This book is an attempt to avoid this problem. Various authors were asked to provide an account of recent developments, techniques, and key examples in their area that would be accessible to others working in nearby fields. The editors asked, in particular, that the authors provide “contributions that illustrated ‘secret handshakes’, yogas and heuristics that experts use privately to guide intuition or simplify calculation but that are replaced by more formal arguments, or simply do not appear, in articles aimed at other specialists.”

The book is addressed specifically to “producers of algebraic geometry”, with the hope that “some consumers from cognate areas” might also profit. For that audience, these three volumes are likely to be of great value.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.

Volume I

Gavril Farkas and Ian Morrison

Logarithmic geometry and moduli
Dan Abramovich, Qile Chen, Danny Gillam, Yuhao Huang, Martin Olsson, Matthew Satriano and Shenghao Sun

Invariant Hilbert schemes
Michel Brion

Algebraic and tropical curves: comparing their moduli spaces
Lucia Caporaso

A superficial working guide to deformations and moduli
F. Catanese

Moduli spaces of hyperbolic surfaces and their Weil–Petersson volumes
Norman Do

Equivariant geometry and the cohomology of the moduli space of curves
Dan Edidin

Tautological and non-tautological cohomology of the moduli space of curves
C. Faber and R. Pandharipande

Alternate compactifications of moduli spaces of curves
Maksym Fedorchuk and David Ishii Smyth

The cohomology of the moduli space of Abelian varieties
Gerard van der Geer

Moduli of K3 surfaces and irreducible symplectic manifolds
V. Gritsenko, K. Hulek and G.K. Sankaran

Normal functions and the geometry of moduli spaces of curves
Richard Hain

Volume II

Parameter spaces of curves
Joe Harris

Global topology of the Hitchin system
Tás Hausel

Differential forms on singular spaces, the minimal model program, and hyperbolicity of moduli stacks
Stefan Kebekus

Contractible extremal rays on \(\mathcal{M}_{0,n}\)
Seán Keel and James McKernan

Moduli of varieties of general type
János Kollár

Singularities of stable varieties
Sándor J Kovács

Soliton equations and the Riemann-Schottky problem
I. Krichever and T. Shiota

GIT and moduli with a twist
Radu Laza

Good degenerations of moduli spaces
Jun Li

Localization in Gromov-Witten theory and Orbifold Gromov-Witten theory
Chiu-Chu Melissa Liu

From WZW models to modular functors
Eduard Looijenga

Shimura varieties and moduli
J.S. Milne

The Torelli locus and special subvarieties
Ben Moonen and Frans Oort

Volume III

Birational geometry for nilpotent orbits
Yoshinori Namikawa

Cell decompositions of moduli space, lattice points and Hurwitz problems
Paul Norbury

Moduli of abelian varieties in mixed and in positive characteristic
Frans Oort

Local models of Shimura varieties, I: Geometry and combinatorics
Georgios Pappas, Michael Rapoport and Brian Smithling

Generalized theta linear series on moduli spaces of vector bundles on curves
Mihnea Popa

Computer aided unirationality proofs of moduli spaces
Frank-Olaf Schreyer

Deformation theory from the point of view of fibered categories
Mattia Talpo and Angelo Vistoli

Mumford’s conjecture — a topological outlook
Ulrike Tillmann

Rational parametrizations of moduli spaces of curves
Alessandro Verra

Hodge loci
Claire Voisin

Homological stability for mapping class groups of surfaces
Nathalie Wahl