Mathematics – Algebraic Geometry
Scientific paper
2002-07-26
Mathematics
Algebraic Geometry
35 pages. To appear in a volume in memory of Paolo Francia, De Gruyter Verlag
Scientific paper
A good canonical projection of a surface $S$ of general type is a morphism to the 3-dimensional projective space P^3 given by 4 sections of the canonical line bundle. To such a projection one associates the direct image sheaf F of the structure sheaf of S, which is shown to admit a certain length 1 symmetrical locally free resolution. The structure theorem gives necessary and sufficient conditions for such a length 1 locally free symmetrical resolution of a sheaf F on P^3 in order that $Spec (mathcal F)$ yield a canonically projected surface of general type. The result was found in 1983 by the first author in the regular case, and the main ingredient here for the irregular (= non Cohen-Macaulay) case is to replace the use of Hilbert resolutions with the use of Beilinson's monads. Most of the paper is devoted then to the analysis of irregular canonical surfaces with p_g=4 and low degree (the analysis in the regular case had been initiated by F. Enriques). For q=1 we classify completely the (irreducible) moduli space for the minimal value K^2=12. We also study other moduli spaces, q=2, K^2=18, and q=3, K^2=12. The last family is provided by the divisors in a (1,1,2) polarization on some Abelian 3-fold. A remarkable subfamily is given by the pull backs, under a 2-isogeny, of the theta divisor of a p.p.Abelian 3-fold: for those surfaces the canonical map is a double cover of a canonical surface (i.e., such that its canonical map is birational). The canonical image is a sextic surface with a plane cubic as double curve, and moreover with an even set of 32 nodes as remaining singular locus. Our method allows to write the equations of these transcendental objects, but not yet to construct new ones by computer algebra.
Catanese Fabrizio
Schreyer Frank Olaf
No associations
LandOfFree
Canonical projections of irregular algebraic surfaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Canonical projections of irregular algebraic surfaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Canonical projections of irregular algebraic surfaces will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-111244