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), 27152732
MSC (2000):
Primary 14J29
Published electronically:
March 19, 2003
MathSciNet review:
1975396
Fulltext 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 CalabiYau threefolds. We prove new results on homogeneous rings associated to a polarized CalabiYau threefold and also prove some existence theorems for CalabiYau covers of threefolds of minimal degree. These have consequences towards constructing new examples of CalabiYau threefolds.
 [Bo]
E.
Bombieri, Canonical models of surfaces of general type, Inst.
Hautes Études Sci. Publ. Math. 42 (1973),
171–219. MR 0318163
(47 #6710)
 [Ca]
F.
Catanese, On the moduli spaces of surfaces of general type, J.
Differential Geom. 19 (1984), no. 2, 483–515.
MR 755236
(86h:14031)
 [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
(86d:14036)
 [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
(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 1665689
(2000a:13023), http://dx.doi.org/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
(2001b:13016)
 [GP2]
Francisco
Javier Gallego and B.
P. Purnaprajna, Very ampleness and higher syzygies for CalabiYau
threefolds, Math. Ann. 312 (1998), no. 1,
133–149. MR 1645954
(99g:14048), http://dx.doi.org/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 Seshadria 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
(84k:14006)
 [HM]
David
W. Hahn and Rick
Miranda, Quadruple covers of algebraic varieties, J. Algebraic
Geom. 8 (1999), no. 1, 1–30. MR 1658196
(99k:14028)
 [H1]
Eiji
Horikawa, Algebraic surfaces of general type with small
𝐶²₁.\ I, Ann. of Math. (2) 104
(1976), no. 2, 357–387. MR 0424831
(54 #12789)
 [H2]
Eiji
Horikawa, Algebraic surfaces of general type with small
𝑐²₁. II, Invent. Math. 37 (1976),
no. 2, 121–155. MR 0460340
(57 #334)
 [H3]
Eiji
Horikawa, Algebraic surfaces of general type with small
𝑐²₁. III, Invent. Math. 47
(1978), no. 3, 209–248. MR 501370
(80h:14012a), http://dx.doi.org/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
(80h:14012b), http://dx.doi.org/10.1007/BF01390285
 [Kod]
Kunihiko
Kodaira, Pluricanonical systems on algebraic surfaces of general
type, J. Math. Soc. Japan 20 (1968), 170–192.
MR
0224613 (37 #212)
 [Kon]
Kazuhiro
Konno, Algebraic surfaces of general type with
𝑐²₁=3𝑝_{𝑔}6, Math. Ann.
290 (1991), no. 1, 77–107. MR 1107664
(92i:14039), http://dx.doi.org/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 (2001h:14049), http://dx.doi.org/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
(44 #209)
 [OP]
Keiji
Oguiso and Thomas
Peternell, On polarized canonical CalabiYau threefolds, Math.
Ann. 301 (1995), no. 2, 237–248. MR 1314586
(96b:14050), http://dx.doi.org/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
(91h:14018), http://dx.doi.org/10.1007/BFb0083344
 [Bo]
 E. Bombieri, Canonical models of surfaces of general type, Inst. Hautes Etudes Sci. Publ. Math. 42 (1973), 171219. MR 47:6710
 [Ca]
 F. Catanese, On the moduli spaces of surfaces of general type, J. Differential Geometry 19 (1984), 483515. 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), 83111. 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), 114. MR 89f:14042
 [GP1]
 F. J. Gallego and B. P. Purnaprajna, Projective normality and syzygies of algebraic surfaces, J. Reine Angew. Math. 506 (1999), 145180. MR 2000a:13023; MR 2001b:13016
 [GP2]
 F. J. Gallego and B. P. Purnaprajna, Very ampleness and higher syzygies for CalabiYau threefolds, Math. Ann. 312 (1998), 133149. 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 Seshadria 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), 10871113. MR 84k:14006
 [HM]
 D. Hahn and R. Miranda, Quadruple covers of algebraic varieties, J. Algebraic Geom. 8 (1999), 130. MR 99k:14028
 [H1]
 E. Horikawa, Algebraic surfaces of general type with small I, Ann. of Math. (2) 104 (1976), 357387. MR 54:12789
 [H2]
 E. Horikawa, Algebraic surfaces of general type with small , II, Invent. Math. 37 (1976), 121155. MR 57:334
 [H3]
 E. Horikawa, Algebraic surfaces of general type with small , III, Invent. Math. 47 (1978), 209248. MR 80h:14012a
 [H4]
 E. Horikawa, Algebraic surfaces of general type with small , IV, Invent. Math. 50 (1978/79), 103128. MR 80h:14012b
 [Kod]
 K. Kodaira, Pluricanonical systems on algebraic surfaces of general type, J. Math. Soc. Japan 20 (1968), 170192. MR 37:212
 [Kon]
 K. Konno, Algebraic surfaces of general type with , Math. Ann. 290 (1991), 77107. MR 92i:14039
 [MP]
 M. Mendes Lopes and R. Pardini, Triple canonical surfaces of minimal degree, International J. Math. 11 (2000), 553578. MR 2001h:14049
 [M]
 D. Mumford, Varieties defined by quadratic equations, Corso CIME in Questions on Algebraic Varieties, Edizioni Cremonese, Rome (1970), 29100. MR 44:209
 [OP]
 K. Oguiso and T. Peternell, On polarized canonical CalabiYau threefolds, Math. Ann. 301 (1995), 237248. MR 96b:14050
 [R]
 M. Reid, Infinitesimal view of extending a hyperplane sectiondeformation theory and computer algebra, Algebraic geometry, Proceedings L'Aquila 1988, 214286, Lecture Notes in Math. 1417, SpringerVerlag, Berlin, 1990. MR 91h:14018
Similar Articles
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 660452142
Email:
purna@math.ukans.edu
DOI:
http://dx.doi.org/10.1090/S0002994703032008
PII:
S 00029947(03)032008
Received by editor(s):
July 5, 2002
Published electronically:
March 19, 2003
Additional Notes:
The first author was partially supported by MCT project number BFM20000621 and by UCM project number PR52/008862. 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
