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Divisor classes associated to families of stable varieties with applications to the moduli space of curves. (English) Zbl 0674.14006

Let \(\pi: X\to T\) be a proper flat morphism of complex algebraic varieties and \(L\) a line bundle on \(X\) with \(X\) and \(T\) separated, \(T\) irreducible and \(X\) pure dimensional. Suppose that \(k=\dim(T)\) and \(d\) is the relative dimension of \(\pi\). Consider the divisor class \({\mathcal E}(L,F)=\pi_*((rc_ 1(L)-\pi^*c_ 1(F))^{d+1}\cap [X]\in A_{k+1}(T)\) for a coherent subsheaf \(F\) of \(\pi_*(L)\). Suppose that the following conditions hold:
(i) If \(t\) is a general point of \(T\), then \(F_ t\otimes C\subset H^ 0(\pi^{-1}(t)\), \(L_{(\pi^{-1}(t))})\) is base point free, very ample and yields a semi-stable embedding of \(\pi^{-1}(t)\).
(ii) \(L\) is relatively ample.
Then, the main result shows
(1) that \({\mathcal E}(L,F)\) lies in the closure of the cone in the \(A_{k- 1}(T)\otimes \mathbb Q\) generated by the effective Weil divisors.
(2) if \(F\) is locally free, \({\mathcal E}(L,F)\) lies in the closure of the cone generated by the effective Cartier divisors.
A counter example due to Ian Morrison shows that the semi-stability condition is essential.
As a significant application of the above result the authors show: Let \(\overline M_ g\) be the moduli space of stable genus \(g\) curves with \(g\geq 2\) and \(\lambda,\delta \in \text{Pic}(\overline M_ g)\otimes\mathbb Q\) be the Hodge and boundary class. Then the class \(a\lambda-b\delta\) has non-negative degree on every curve in \(\overline M_ g\) not contained in \(\Delta =\overline M_ g- M_ g\) if and only if \(a\geq (8+4/g)b\). Furthermore, \(a\lambda-b\delta\) is ample if and only if \(a>11.b>0\). These results are known in part.

MSC:

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

References:

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