Lectures on moduli of curves. Notes by D. R. Gokhale (Q790193)

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 [\textit{D. Mumford}, Enseign. Math., II. Sér. 23, 39-110 (1977; Zbl 0363.14003)].