On the bicanonical morphism of quadruple Galois canonical covers

Gallego, Francisco; Purnaprajna, Bangere

doi:10.1090/S0002-9947-2011-05353-5

On the bicanonical morphism of quadruple Galois canonical covers
HTML articles powered by AMS MathViewer

by Francisco Javier Gallego and Bangere P. Purnaprajna

Trans. Amer. Math. Soc. 363 (2011), 4401-4420

DOI: https://doi.org/10.1090/S0002-9947-2011-05353-5

Published electronically: March 7, 2011

PDF | Request permission

Abstract:

In this article we study the bicanonical map $\varphi _2$ of quadruple Galois canonical covers $X$ of surfaces of minimal degree. We show that $\varphi _2$ has diverse behavior and exhibits most of the complexities that are possible for a bicanonical map of surfaces of general type, depending on the type of $X$. There are cases in which $\varphi _2$ is an embedding, and if it so happens, $\varphi _2$ embeds $X$ as a projectively normal variety, and there are cases in which $\varphi _2$ is not an embedding. If the latter, $\varphi _2$ is finite of degree $1$, $2$ or $4$. We also study the canonical ring of $X$, proving that it is generated in degree less than or equal to $3$ and finding the number of generators in each degree. For generators of degree $2$ we find a nice general formula which holds for canonical covers of arbitrary degrees. We show that this formula depends only on the geometric and the arithmetic genus of $X$.

References

Lucian Bădescu, Algebraic surfaces, Universitext, Springer-Verlag, New York, 2001. Translated from the 1981 Romanian original by Vladimir Maşek and revised by the author. MR 1805816, DOI 10.1007/978-1-4757-3512-3
Mauro C. Beltrametti and Tomasz Szemberg, On higher order embeddings of Calabi-Yau threefolds, Arch. Math. (Basel) 74 (2000), no. 3, 221–225. MR 1739501, DOI 10.1007/s000130050434
G. Casnati, Examples of Calabi-Yau threefolds as covers of almost-Fano threefolds, Geom. Dedicata 119 (2006), 169–179. MR 2247656, DOI 10.1007/s10711-006-9067-y
Francisco Javier Gallego and B. P. Purnaprajna, Very ampleness and higher syzygies for Calabi-Yau threefolds, Math. Ann. 312 (1998), no. 1, 133–149. MR 1645954, DOI 10.1007/s002080050215
Francisco Javier Gallego and B. P. Purnaprajna, Some results on rational surfaces and Fano varieties, J. Reine Angew. Math. 538 (2001), 25–55. MR 1855753, DOI 10.1515/crll.2001.068
Francisco Javier Gallego and Bangere P. Purnaprajna, On the canonical rings of covers of surfaces of minimal degree, Trans. Amer. Math. Soc. 355 (2003), no. 7, 2715–2732. MR 1975396, DOI 10.1090/S0002-9947-03-03200-8
Francisco Javier Gallego and Bangere P. Purnaprajna, Classification of quadruple Galois canonical covers. II, J. Algebra 312 (2007), no. 2, 798–828. MR 2333185, DOI 10.1016/j.jalgebra.2006.11.011
Francisco Javier Gallego and Bangere P. Purnaprajna, Classification of quadruple Galois canonical covers. I, Trans. Amer. Math. Soc. 360 (2008), no. 10, 5489–5507. MR 2415082, DOI 10.1090/S0002-9947-08-04587-X
Mark L. Green, The canonical ring of a variety of general type, Duke Math. J. 49 (1982), no. 4, 1087–1113. MR 683012
Eiji Horikawa, Algebraic surfaces of general type with small $C^{2}_{1}.$ I, Ann. of Math. (2) 104 (1976), no. 2, 357–387. MR 424831, DOI 10.2307/1971050
Kazuhiro Konno, Algebraic surfaces of general type with $c^2_1=3p_g-6$, Math. Ann. 290 (1991), no. 1, 77–107. MR 1107664, DOI 10.1007/BF01459239
Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279. MR 276239
David Mumford, Varieties defined by quadratic equations, Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969) Edizioni Cremonese, Rome, 1970, pp. 29–100. MR 0282975
Keiji Oguiso and Thomas Peternell, On polarized canonical Calabi-Yau threefolds, Math. Ann. 301 (1995), no. 2, 237–248. MR 1314586, DOI 10.1007/BF01446628
Gang Xiao, Degree of the bicanonical map of a surface of general type, Amer. J. Math. 112 (1990), no. 5, 713–736. MR 1073006, DOI 10.2307/2374804