Lectures on moduli of curves. Notes by D. R. Gokhale. (English) Zbl 0534.14012

The main result of this book is (roughly): a curve is stable if and only if it can be stably embedded in a canonical way (i.e. it is stable in the sense of geometric invariant theory à la Mumford). The notion of a stable curve was introduced by Mumford et al. as a reduced curve with only double points and without continuous automorphisms (in the case of genus \(\geq 2)\) (if the latter condition is omitted, we obtain the notion of a semistable curve) to construct a canonical compactification of the moduli space of smooth curves.
More precisely there are the following two theorems: (I) (Theorem 1.0.1). Let C be a connected (but possibly reducible) curve in \({\mathbb{P}}^ n\), and consider its m-th Hilbert point (the point in the Hilbert scheme corresponding to the embedding obtained by composing \(C\subset {\mathbb{P}}^ n\) with the Veronese map \({\mathbb{P}}^ n\subset {\mathbb{P}}^ N, N=\left( \begin{matrix} n+m\\ n\end{matrix} \right))\). Then if the m-th Hilbert point is semistable, then C is semi-stable. - (II) (Theorem 2.0.2). By taking the quotient, we obtain the moduli space of stable curves which turns out to be irreducible and projective. As a corollary of the second result we have: the n-canonical embedding (embedding by n-th tensor of the dualizing (canonical) sheaf) of a stable curve is stable if \(n\geq 10.\)
The second result was first obtained by Knudsen by a different method, and by Mumford by a similar method but with the Chow scheme [D. Mumford, Enseign. Math., II. Sér. 23, 39-110 (1977; Zbl 0363.14003)].
Reviewer: Y.Namikawa


14H10 Families, moduli of curves (algebraic)
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14C05 Parametrization (Chow and Hilbert schemes)
14D20 Algebraic moduli problems, moduli of vector bundles
14D22 Fine and coarse moduli spaces


Zbl 0363.14003