This article settles the question of existence of smooth weak Fano threefolds of Picard number two with small anti-canonical map and previously classified numerical invariants obtained by blowing up certain curves on smooth Fano threefolds of Picard number 1 with the exception of 12 numerical cases.

We study a family of semiample divisors on the moduli space M0,n that come from the theory of conformal blocks for the Lie algebra sln and level 1. The divisors we study are invariant under the action of Sn on M0,n. We compute their classes and prove that they generate extremal rays in the cone of symmetric nef divisors on M0,n. In particular, these divisors define birational contractions of M0,n, which we show factor through reduction morphisms to moduli spaces of weighted pointed curves defined by Hassett.

It was conjectured in [Nam73] that the Torelli map Mg → Ag associating to a curve its jacobian extends to a regular map from the Deligne-Mumford moduli space of stable curves Mg to the (normalization of the) Igusa blowup A cent g . A counterexample in genus g = 9 was found in [AB11]. Here, we prove that the extended map is regular for all g ≤ 8, thus completely solving the problem in every genus.Date: May 14, 2011.

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