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

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

Published electronically:
March 19, 2003

MathSciNet review:
1975396

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 Etudes Sci. Publ. Math.**42**(1973), 171-219. MR**47:6710****[Ca]**F. Catanese,*On the moduli spaces of surfaces of general type*, J. Differential Geometry**19**(1984), 483-515. MR**86h:14031****[Ci]**C. Ciliberto,*Sul grado dei generatori dell'anello di una superficie di tipo generale*, Rend. Sem. Mat. Univ. Politec. Torino**41**(1983), 83-111. MR**86d:14036****[EH]**D. Eisenbud and J. Harris,*On varieties of minimal degree (a centennial account)*, Algebraic Geometry, Bowdoin 1985, Amer. Math. Soc. Sympos. in Pure and Appl. Math.**46**(1987), 1-14. MR**89f:14042****[GP1]**F. J. Gallego and B. P. Purnaprajna,*Projective normality and syzygies of algebraic surfaces*, J. Reine Angew. Math.**506**(1999), 145-180. MR**2000a:13023**; MR**2001b:13016****[GP2]**F. J. Gallego and B. P. Purnaprajna,*Very ampleness and higher syzygies for Calabi-Yau threefolds*, Math. Ann.**312**(1998), 133-149. MR**99g:14048****[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]**M. L. Green,*The canonical ring of a variety of general type*, Duke Math. J.**49**(1982), 1087-1113. MR**84k:14006****[HM]**D. Hahn and R. Miranda,*Quadruple covers of algebraic varieties*, J. Algebraic Geom.**8**(1999), 1-30. MR**99k:14028****[H1]**E. Horikawa,*Algebraic surfaces of general type with small**I*, Ann. of Math. (2)**104**(1976), 357-387. MR**54:12789****[H2]**E. Horikawa,*Algebraic surfaces of general type with small**, II*, Invent. Math.**37**(1976), 121-155. MR**57:334****[H3]**E. Horikawa,*Algebraic surfaces of general type with small**, III*, Invent. Math.**47**(1978), 209-248. MR**80h:14012a****[H4]**E. Horikawa,*Algebraic surfaces of general type with small**, IV*, Invent. Math.**50**(1978/79), 103-128. MR**80h:14012b****[Kod]**K. Kodaira,*Pluricanonical systems on algebraic surfaces of general type*, J. Math. Soc. Japan**20**(1968), 170-192. MR**37:212****[Kon]**K. Konno,*Algebraic surfaces of general type with*, Math. Ann.**290**(1991), 77-107. MR**92i:14039****[MP]**M. Mendes Lopes and R. Pardini,*Triple canonical surfaces of minimal degree*, International J. Math.**11**(2000), 553-578. MR**2001h:14049****[M]**D. Mumford,*Varieties defined by quadratic equations*, Corso CIME in Questions on Algebraic Varieties, Edizioni Cremonese, Rome (1970), 29-100. MR**44:209****[OP]**K. Oguiso and T. Peternell,*On polarized canonical Calabi-Yau threefolds*, Math. Ann.**301**(1995), 237-248. MR**96b:14050****[R]**M. Reid,*Infinitesimal view of extending a hyperplane section--deformation theory and computer algebra*, Algebraic geometry, Proceedings L'Aquila 1988, 214-286, Lecture Notes in Math.**1417**, Springer-Verlag, Berlin, 1990. MR**91h:14018**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
14J29

Retrieve articles in all journals with MSC (2000): 14J29

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