### Impacto

Gallego Rodrigo, Francisco Javier and Purnaprajna, Bangere P.
(1996)
*Normal presentation on elliptic ruled surfaces.*
Journal of Algebra, 186
(2).
pp. 597-625.
ISSN 0021-8693

PDF
Restringido a Repository staff only hasta 2020. 287kB |

Official URL: http://www.sciencedirect.com/science/article/pii/S0021869396903889

## Abstract

From the introduction: Let X be an irreducible projective variety and L a very ample lLine bundle on X, whose complete linear series defines 'L : X ! P(H0(L)). Let S = 1

m=0 SmH0(X,L) and let R(L) = L1 n=0 H0(X,L n) be the homogeneous coordinate ring associated to L. Then R is a finitely generated graded module over S, so it has a

minimal graded free resolution. We say that the line bundle L is normally generated if the natural maps SmH0(X,L) ! H0(X,L m) are surjective for all m 2. If L is

normally generated, then we say that L satisfies property Np, if the matrices in the free resolution of R over S have linear entries until the p-th stage. In particular, property

N1 says that the homogeneous ideal I of X in P(H0(L)) is generated by quadrics. A line bundle satisfying property N1 is also called normally presented. Let R = kR1R2. . .

be a graded algebra over a field k. The algebra R is a Koszul ring iff TorRi (k, k) has pure degree i for all i. In this article we determine exactly (theorem 4.2) which line bundles on an elliptic ruled surface X are normally presented. As a corollary we show that Mukai’s conjecture is true for the normal presentation of the adjoint linear series for an elliptic ruled surface.

In section 5 of this article, we show that if L is normally presented on X then the homogeneous coordinate ring associated to L is Koszul. We also give a new proof of

the following result due to Butler: If deg(L) 2g + 2 on a curve X of genus g, then L embeds X with Koszul homogeneous coordinate ring.

Item Type: | Article |
---|---|

Uncontrolled Keywords: | Normal presentation of line bundles; Elliptic ruled surface; Mukai’s conjecture; Adjoint linear series; homogeneous coordinate ring |

Subjects: | Sciences > Mathematics > Algebraic geometry |

ID Code: | 15418 |

References: | J. Backelin and R. Fr¨oberg, Veronese subrings, and rings with linear resolutions,Re¨. Roumaine Math. Pures Appl. 30 1985., 85]97. D. Butler, Normal generation of vector bundles over a curve, J. Differential Geom. 39 1994., 1]34. G. Castelnuovo, Sui multipli di uni serie di gruppi di punti appatente ad una curva algebraica, Rend. Circ. Mat. Palermo 7 1892., 99]119. T. Fujita, Defining equations for certain types of polarized varieties, in ‘‘Complex Analysis and Algebraic Geometry,’’ pp. 165]173, Cambridge Univ. Press, Cambridge,UK, 1977. F. J. Gallego and B. P. Purnaprajna, Higher syzygies of elliptic ruled surfaces, J. Algebra 186 1996.. M. Green, Koszul cohomology and the geometry of projective varieties,J. Differen-tial Geom. 19 1984., 125]171. M. Green and R. Lazarsfeld, Some results on the syzygies of finite sets and algebraic curves, Compositio Math. 67 1989., 301]314. R. Hartshorne, ‘‘Algebraic Geometry,’’ Springer-Verlag, Berlin, 1977. Y. Homma, Projective normality and the defining equations of ample invertible sheaves on elliptic ruled surfaces with eG0, Natur. Sci. Rep. Ochanomizu Uni¨. 31 1980., 61]73. Y. Homma, Projective normality and the defining equations of an elliptic ruled surface with negative invariant, Natur. Sci. Rep. Ochanomizu Uni¨. 33 1982., 17]26. Y. Miyaoka, The Chern class and Kodaira dimension of a minimal variety, in ‘‘Algebraic Geometry}Sendai 1985,’’ Advanced Studies in Pure Math., Vol. 10, pp.449]476, North-Holland, Amsterdam. D. Mumford, Varieties defined by quadratic equations, in ‘‘Corso CIME in Questions on Algebraic Varieties, Rome, 1970,’’ pp. 30]100. G. Pareschi, Koszul algebras associated to adjunction bundles, J. Algebra 157 1993., 161]169. A. Polishchuk, On the Koszul property of the homogeneous coordinate ring of a curve, J. Algebra 178 1995., 122]135. I. Reider, Vector bundles of rank 2 and linear systems on an algebraic surface,Ann. of Math. (2) 127 1988., 309]316. B. St.-Donat, Sur les equations d´efinissant une courbe alg´ebrique, C.R. Acad. Sci. Paris S´er. I Math. 274 1972., 324]327 |

Deposited On: | 30 May 2012 08:13 |

Last Modified: | 06 Feb 2014 10:24 |

Repository Staff Only: item control page