On the canonical rings of covers of surfaces of minimal degree

Authors:
Francisco Javier Gallego and Bangere P. Purnaprajna

Journal:
Trans. Amer. Math. Soc. **355** (2003), 2715-2732

MSC (2000):
Primary 14J29

Published electronically:
March 19, 2003

MathSciNet review:
1975396

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Abstract: In one of the main results of this paper, we find the degrees of the generators of the canonical ring of a regular algebraic surface of general type defined over a field of characteristic , under the hypothesis that the canonical divisor of determines a morphism from to a surface of minimal degree . As a corollary of our results and results of Ciliberto and Green, we obtain a necessary and sufficient condition for the canonical ring of to be generated in degree less than or equal to . We construct new examples of surfaces satisfying the hypothesis of our theorem and prove results which show that many a priori plausible examples cannot exist. Our methods are to exploit the -algebra structure on . These methods have other applications, including those on Calabi-Yau threefolds. We prove new results on homogeneous rings associated to a polarized Calabi-Yau threefold and also prove some existence theorems for Calabi-Yau covers of threefolds of minimal degree. These have consequences towards constructing new examples of Calabi-Yau threefolds.

**[Bo]**E. Bombieri,*Canonical models of surfaces of general type*, Inst. Hautes Études Sci. Publ. Math.**42**(1973), 171–219. MR**0318163****[Ca]**F. Catanese,*On the moduli spaces of surfaces of general type*, J. Differential Geom.**19**(1984), no. 2, 483–515. MR**755236****[Ci]**Ciro Ciliberto,*The degree of the generators of the canonical ring of a surface of general type*, Rend. Sem. Mat. Univ. Politec. Torino**41**(1983), no. 3, 83–111 (1984) (Italian, with English summary). MR**778862****[EH]**David Eisenbud and Joe Harris,*On varieties of minimal degree (a centennial account)*, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 3–13. MR**927946**, 10.1090/pspum/046.1/927946**[GP1]**F. J. Gallego and B. P. Purnaprajna,*Projective normality and syzygies of algebraic surfaces*, J. Reine Angew. Math.**506**(1999), 145–180. MR**1665689**, 10.1515/crll.1999.506.145

F. J. Gallego and B. P. Purnaprajna,*Erratum to the paper: “Projective normality and syzygies of algebraic surfaces” [J. Reine Angew. Math. 506 (1999), 145–180; MR1665689 (2000a:13023)]*, J. Reine Angew. Math.**523**(2000), 233–234. MR**1762962****[GP2]**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**, 10.1007/s002080050215**[GP3]**F. J. Gallego and B. P. Purnaprajna,*Canonical covers of varieties of minimal degree*, Preprint math.AG/0205010. To appear in ``A tribute to Seshadri--a collection of papers on Geometry and Representation Theory'', Hindustan Book Agency (India) Ltd.**[GP4]**F. J. Gallego and B. P. Purnaprajna,*Some homogeneous rings associated to finite morphisms*, Preprint. To appear in ``Advances in Algebra and Geometry'' (Hyderabad Conference 2001), Hindustan Book Agency (India) Ltd.**[GP5]**F. J. Gallego and B. P. Purnaprajna,*On the rings of trigonal curves*, in preparation.**[G]**Mark L. Green,*The canonical ring of a variety of general type*, Duke Math. J.**49**(1982), no. 4, 1087–1113. MR**683012****[HM]**David W. Hahn and Rick Miranda,*Quadruple covers of algebraic varieties*, J. Algebraic Geom.**8**(1999), no. 1, 1–30. MR**1658196****[H1]**Eiji Horikawa,*Algebraic surfaces of general type with small 𝐶²₁. I*, Ann. of Math. (2)**104**(1976), no. 2, 357–387. MR**0424831****[H2]**Eiji Horikawa,*Algebraic surfaces of general type with small 𝑐²₁. II*, Invent. Math.**37**(1976), no. 2, 121–155. MR**0460340****[H3]**Eiji Horikawa,*Algebraic surfaces of general type with small 𝑐²₁. III*, Invent. Math.**47**(1978), no. 3, 209–248. MR**501370**, 10.1007/BF01579212**[H4]**Eiji Horikawa,*Algebraic surfaces of general type with small 𝑐²₁. IV*, Invent. Math.**50**(1978/79), no. 2, 103–128. MR**517773**, 10.1007/BF01390285**[Kod]**Kunihiko Kodaira,*Pluricanonical systems on algebraic surfaces of general type*, J. Math. Soc. Japan**20**(1968), 170–192. MR**0224613****[Kon]**Kazuhiro Konno,*Algebraic surfaces of general type with 𝑐²₁=3𝑝_{𝑔}-6*, Math. Ann.**290**(1991), no. 1, 77–107. MR**1107664**, 10.1007/BF01459239**[MP]**Margarida Mendes Lopes and Rita Pardini,*Triple canonical surfaces of minimal degree*, Internat. J. Math.**11**(2000), no. 4, 553–578. MR**1768173**, 10.1142/S0129167X00000271**[M]**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****[OP]**Keiji Oguiso and Thomas Peternell,*On polarized canonical Calabi-Yau threefolds*, Math. Ann.**301**(1995), no. 2, 237–248. MR**1314586**, 10.1007/BF01446628**[R]**Miles Reid,*Infinitesimal view of extending a hyperplane section—deformation theory and computer algebra*, Algebraic geometry (L’Aquila, 1988) Lecture Notes in Math., vol. 1417, Springer, Berlin, 1990, pp. 214–286. MR**1040562**, 10.1007/BFb0083344

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Additional Information

**Francisco Javier Gallego**

Affiliation:
Departamento de Álgebra, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain

Email:
FJavier_Gallego@mat.ucm.es

**Bangere P. Purnaprajna**

Affiliation:
Department of Mathematics, University of Kansas, 405 Snow Hall, Lawrence, Kansas 66045-2142

Email:
purna@math.ukans.edu

DOI:
https://doi.org/10.1090/S0002-9947-03-03200-8

Received by editor(s):
July 5, 2002

Published electronically:
March 19, 2003

Additional Notes:
The first author was partially supported by MCT project number BFM2000-0621 and by UCM project number PR52/00-8862. The second author was partially supported by the General Research Fund of the University of Kansas at Lawrence. The first author is grateful for the hospitality of the Department of Mathematics of the University of Kansas at Lawrence.

Article copyright:
© Copyright 2003
American Mathematical Society