On the Classification of Threefolds Isogenous to a Product
Christian Gleissner (Bayreuth)
Aula Beltrami - Thursday, April 28, 2016 h.16:00
Abstract. A complex projective variety X is said to be isogenous to a product if X is a quotient
X = (C1 × ... × Cn)/G,
where the Ci where the Ci ’s are curves of genus at least two, and G is a finite group acting freely on
C1x...xCn.
In the surface case n = 2 these varieties, of general type, has been studied
extensively. For example, the full classification for χ(OX) = 1 is available thanks to
[BCG08], [CP09], [Pe10] et al. However, relatively view is known in higher dimensions.
In this talk we consider the threefold case i.e. n = 3. We give an outline of an algorithm
to classify these varieties for a fixed value of χ(OX) and present, as an application,
the classification in the boundary case χ(OX) = −1, which was (partially) obtained in
a joint work with Davide Frapporti [FG14]. Our methodology is a blend of algebraic
geometry, group theory and computer algebra.
References
[BCG08] I. Bauer, F. Catanese, F. Grunewald, The classification of surfaces with pg = q = 0 isogenus to a product. Pure Appl. Math. Q., 4, no.2, part1, (2008), 547–586.
[Cat00] F. Catanese, Fibred surfaces, varieties isogenus to a product and related moduli spaces. Amer. J. Math., 122, (2000), 1–44.
[CP09] G. Carnovale, F. Polizzi, The classification of surfaces with pg = q = 1 isogenus to a product of curves. Adv. Geom., 9, no.2, (2009), 233–256.
[FG14] D. Frapporti, C. Gleißner, On threefolds isogenous to a product of curves. arXiv: 1412.6365v2, (2014).
[Pe10] M. Penegini, The Classification of Isotrivially Fibred Surfaces with pg = q = 2, and topics on Beauville Surfaces. PhD thesis, Universit ̈at Bayreuth, (2010).
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