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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

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14C20 Divisors, linear systems, invertible sheaves 14D20 Algebraic moduli problems, moduli of vector bundles
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##### References:
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