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The Kodaira dimension of the moduli space of curves of genus \(\geq 23\). (English) Zbl 0631.14023
Here it is proved that the moduli scheme \(\bar M_ g\) of genus-g stable curves is of general type if \(g>23\) and has Kodaira dimension \(>0\) if \(g=23\), refining the results and simplifying the proofs of the fundamental paper by J. Harris and D. Mumford in Invent. Math. 67, 23-86 (1982; Zbl 0506.14016)]. The proofs here use the theory of limit linear series on stable curves with \(Pic^ 0\) compact introduced by the authors in Invent. Math. 85, 337-371 (1986; Zbl 0598.14003) and applied by the authors (and others) to obtain several interesting results. The proof uses the computation in \(Pic(M_ g)\otimes {\mathbb{Q}}\) of the class of the following divisors: \(D^ r_ s\) (when \(g+1\) is not prime, \(g+1=(r+1)(s-1))\), \(E_ s\) (when g is even, \(g=2(s-1))\); \(E_ s\) is the closure of the set of genus \(g\) curves with a linear series of degree \(s\) and dimension 1 violating Petri’s condition; \(D^ r_ s\) is the closure of the locus of curves with a \(g^ r_ d\) with \(d=rs-1\). Furthermore, when \(g=(r+1)(s-1)-1\), on a general curve C, \(W^ r_ d(C)\) is a smooth, irreducible curve; the authors need and compute its genus.
Reviewer: E.Ballico

14H10 Families, moduli of curves (algebraic)
14C20 Divisors, linear systems, invertible sheaves
14D20 Algebraic moduli problems, moduli of vector bundles
Full Text: DOI EuDML
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