## 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

### MSC:

 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

### Keywords:

Hilbert scheme; stable point; moduli space of stable curves

Zbl 0363.14003