Finite commutative rings and their applications / Gilberto Bini, Flaminio Flamini. p.cm.--(Kluwer international series in engineering and computer science; SECS 680) lncludes bibliographical references and index.

Abstract. Let M n g be the moduli space of n-pointed Riemann surfaces of genus g. Denote by M n g the Deligne-Mumford compactification of M n g . In the present paper, we calculate the orbifold and the ordinary Euler characteristics of M n g for any g and n such that n > 2 − 2g.

We compute the Kodaira dimension of the universal Picard variety P d,g parameterizing line bundles of degree d on curves of genus g under the assumption that (d − g + 1, 2g − 2) = 1. We also give partial results for arbitrary degrees d and we investigate for which degrees the universal Picard varieties are birational.1991 Mathematics Subject Classification. 14H10. -Geometria delle varietà algebriche e dei loro spazi di moduli. 1 2 GILBERTO BINI, CLAUDIO FONTANARI, FILIPPO VIVIANI since P d,g is singular and not projective, the Kodaira dimension of P d,g , which we denote by κ(P d,g ), is defined as the Kodaira dimension of any smooth projective model of it (see [Laz04, Example 2.1.5]). The previous result of Verra implies that κ(P d,g ) = −∞ for g ≤ 9 and any d.Theorem 1.2. Assume that (d − g + 1, 2g − 2) = 1 and g ≥ 10. The Kodaira dimension of P d,g is equal toIn Propositions 6.5 and 6.3, we also determine the Iitaka fibration (see [Laz04, Def. 1.3.6]) of P d,g in the non-trivial cases, namely for g ≥ 11. Without any assumption on the degree d, we obtain the following partial result:Theorem 1.3. The Kodaira dimension of P d,g (for g ≥ 10) satisfies the following inequalitiesMoreover, κ(P d,g ) = 3g −3 if κ(M g ) ≥ 0 (and in particular for g ≥ 22).Let us now explain the strategy that we use to prove the above results. The main tool we use is the GIT compactification constructed by Caporaso (see [Cap94]) φ d : P d,g → M g of P d,g over the Deligne-Mumford moduli space M g of stable curves of genus g. The projective normal variety P d,g is a good moduli space for the stack Pic d,g (see [Cap08] and [Mel09]), whose section over a scheme S is the groupoid Pic d,g (S) of families of quasistable curves of genus g f : (C, L) → S endowed with a balanced line bundle L of degree d (see 2.1 for details). Furthermore, P d,g is a coarse moduli scheme for Pic d,g if and only if (d − g + 1, 2g − 2) = 1, which is precisely the numerical hypothesis on the degree d in Theorem 1.2.Albeit P d,g is singular, we can prove (under the same assumption on the degree) that P d,g has canonical singularities and therefore pluricanonical forms on the smooth locus lift to any desingularization:Theorem 1.4. Assume that (d−g+1, 2g−2) = 1 and that g ≥ 4. Then P d,g has canonical singularities. In particular, if P d,g is a resolution ON THE BIRATIONAL GEOMETRY OF P d,g 3 of singularities of P d,g , then every pluricanonical form defined on the smooth locus P reg d,g of P d,g extends holomorphically to P d,g , that is, for all integers m we have h 0 (P reg d,g , mK P reg d,g ) = h 0 ( P d,g , mK P d,g ). The proof of this theorem is given in Section 4. The restriction on the degree d comes from the fact that P d,g has finite quotient singularities if and only if (d − g + 1, 2g − 2) = 1; hence only for such degrees d we can apply the Reid-Tai criterion for the canonicity of finite quotient singularities (see e.g. [HM82, p. 27] or [Lud07, Thm. 4.1.11]). Indeed, we establish in Theorem 4.8 a similar statement without any restriction on d for the open ...

In this paper, we investigate quotients of Calabi-Yau manifolds Y embedded in Fano varieties X, which are products of two del Pezzo surfaces -with respect to groups G that act freely on Y . In particular, we revisit some known examples and we obtain some new Calabi-Yau varieties with small Hodge numbers. The groups G are subgroups of the automorphism groups of X, which is described in terms of the automorphism group of the two del Pezzo surfaces. e-print archive: http://lanl.arXiv.org/abs/1104.0247v1