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Torelli theorem for stable curves. (English) Zbl 1230.14037

The authors study the generalization of the classical Torelli morphism from the moduli of stable curves to the moduli of principally polarized stable semi-abelic pairs. The main results are concerned with the study of the fibres and of its injectivity locus.
Let us be more precise. The classical result of R. Torelli [“Sulle varietà di Jacobi”, Rom. Acc. L. Rend. (5) 22, No. 2, 98–103, 437–441 (1913; JFM 44.0655.03)] claims the injectivity of the map from the moduli scheme of smooth projective curves of genus \(g\), \(M_g\), to the moduli scheme of principally polarized abelian varieties of dimension \(g\), \(A_g\), that maps a curve to its jacobian variety together with the theta divisor. The most common compactification of \(M_g\) is the moduli space of Deligne-Mumford stable curves, \(\bar M_g\). Thus, one is naturally concerned with the study of those compactifications of \(A_g\) together with a map from \(\bar M_g\) to it whose restriction to \(M_g\) coincides with the Torelli map.
An instance of such a pair is given by the second Voronoi toroidal compactification of \(A_g\) together with a suitable map. However, this map fails to be injective and, indeed, may have positive-dimensional fibers. We refer the reader to [Y. Namikawa, Toroidal compactification of Siegel spaces. Lecture Notes in Mathematics. 812. Berlin-Heidelberg-New York: Springer-Verlag (1980; Zbl 0466.14011)] and [V. Vologodsky, The extended Torelli and Prym maps. Univ. of Georgia PhD thesis (2003)].
In this paper, the authors deal with a second instance. Namely, they consider the coarse moduli space of principally polarized semi-abelic stable pairs introduced by Alexeev as well as an extension of the Torelli map for this case [V. Alexeev, “Complete moduli in the presence of semiabelian group action”, Ann. Math. (2) 155, No. 3, 611–708 (2002; Zbl 1052.14017); “Compactified Jacobians and Torelli map”, Publ. Res. Inst. Math. Sci. 40, No. 4, 1241–1265 (2004; Zbl 1079.14019)].
The first main result is that the compactified Torelli map is injective at curves having \(3\)-edge-connected dual graph (e.g. irreducible curves, curves with two components intersecting in at least three points). The second one offers different charactizations of curves having the same image by the compactified Torelli map.

MSC:

14H40 Jacobians, Prym varieties
14H51 Special divisors on curves (gonality, Brill-Noether theory)
14K30 Picard schemes, higher Jacobians
14D20 Algebraic moduli problems, moduli of vector bundles
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References:

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