On the normality of the rings of Schubert varieties

We prove that the cone over a Schubert variety inG/P (P being a maximal parabolic subgroup of classical type) is normal by exhibiting a 2-regular sequence inR(w) (the homogeneous coordinate ring of the Schubert varietyX(w) inG/P under the canonical protective embeddingG/P ⊂→ (p (H° G/P,L)),L being the ample generator of (PicG/P), which vanishes on the singular locus ofX(w). We also prove the surjectivity ofH° (G/Q, L) H° (X(w), L), whereQ is a classical parabolic subgroup (not necessarily maximal) ofG andL is an ample line bundle onG/Q.     

Very Ampleness of Line Bundles and Canonical Embedding of Coverings of Manifolds

Let L be an ample line bundle on a Kähler manifolds of nonpositive sectional curvature with K as the canonical line bundle. We give an estimate of m such that K+mL is very ample in terms of the injectivity radius. This implies that m can be chosen arbitrarily small once we go deep enough into a tower of covering of the manifold. The same argument gives an effective Kodaira Embedding Theorem for compact Kähler manifolds in terms of sectional curvature and the injectivity radius. In case of locally Hermitian symmetric space of noncompact type or if the sectional curvature is strictly negative, we prove that K itself is very ample on a large covering of the manifold.     

Ruled threefolds homeomorphic to ℙ2 × ℙ1 and their divisors

We study a class of ruled threefolds, namely, manifolds X with a projection p:X→ℙ², such that each fiber is isomorphic to ℙ¹, and which are homeomorphic to ℙ²×ℙ¹; and we characterize ample and very ample line bundles on such threefolds. This paper was written with the financial support of M.P.I. The author is a member of G.N.S.A.G.A. of the C.N.R.     

Generation of k-jets on toric varieties

The notion of a k-convex -support function for a toric variety is introduced. A criterion for a line bundle L to generate k-jets on X is given in terms of the k-convexity of the -support function . Equivalently L is proved to be k-jet ample if and only if the restriction to each invariant curve has degree at least k. Received October 22, 1997; in final form January 12, 1998     

Automorphisms and moduli spaces of varieties with ample canonical class via deformations of abelian covers

By a recent result of Viehweg, projective manifolds with ample canonical class have a coarse moduli space, which is a union of quasiprojective varieties.In this paper, we prove that there are manifolds with ample canonical class that lie on arbitrarily many irreducible components of the moduli; moreover, for any finite abelian group G there exist infinitely many components M of the moduli of varieties with ample canonical class such that the generic automorphism group G^Mis equal to G. In order to construct the examples, we use abelian covers. Let Y be a smooth complex projective variety of dimension ? 2. A Galois cover f :X ? Y whose Galois group is finite and abelian is called an abelian cover of Y; by [Pal], it is determined by its building data, i.e. by the branch divisors and by some line bundles on Y, satisfying appropriate compatibility conditions. Natural deformations of an abelian cover are also introduced in [Pal]. In this paper we prove two results about abelian covers:first, that if the building data are sufficiently ample, then the natural deformations surject on the Kuranishi family of X; second, that if the building data are sufficiently ample and generic, then Aut(X)= G.     

Projective surfaces withk-very ample line bundles of genus ≤3k+1

Let be the set of surfaces,S, polarized by a k-very ample line bundle,L, with genus≤3k+1. All the elements (S, L) of are listed. The classification of surfaces polarized by ak-very ample line bundle of degree ≤4k+4 is completed by proving that this class of surfaces is a subset of .     

Ample vector bundles and Bordiga surfaces

Let X be a smooth complex projective variety and let Z ? X be a smooth surface, which is the zero locus of a section of an ample vector bundle ? of rank dimX – 2 ≥ 2 on X. Let H be an ample line bundle on X, whose restriction H _Z to Z is a very ample line bundle and assume that (Z, H _Z ) is a Bordiga surface, i.e., a rational surface having (?², ?? (4)) as its minimal adjunction theoretic reduction. Triplets (X, ?, H) as above are discussed and classified. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dimensions of very ample pluricanonical subsystems for compact quotients of bounded symmetric domains

A compact complex manifold X obtained by taking quotient of a bounded symmetric domain has an ample canonical line bundle. We prove that the dimension of very ample pluricanonical subsystem is strictly bigger than 2n, where n is the dimension of X. Received: 23 June 2000 / Revised version: 30 March 2001     

k—Very Ample Line Bundles on Del Pezzo Surfaces

A k-very ample line bundle L on a Del Pezzo Surface is numerically characterized, improving the results of Biancofiore— Ceresa in [7].     

Projective normality of abelian surfaces of type (1, 2<Emphasis Type="Italic">d</Emphasis>)

We show that an abelian surface embedded in P^N by a very ample line bundle of type (1,2d) is projectively normal if and only if d4. This completes the study of the projective normality of abelian surfaces embedded by complete linear systems.Supported by EAGER.Mathematics Subject Classification (2000): Primary, 14K05; Secondary, 14N05, 14E20     

On the 2 very ampleness of the adjoint bundle

Let S be a smooth projective surface over C polarized by a 2-very ample line bundle L=O _S(L), i.e. for any 0-dimensional subscheme (Z,O _Z ) of length 3 the restriction map Γ(L)→Γ(L⊗O _Z) is a surjection. This generalization of very ampleness was recently introduced by M. Beltrametti and A.J. Sommese. The authors prove that, if L·L≥13, the adjoint line bundleK _S⊗L is 2-very ample apart from a list of well understood exceptions and up to contracting down the smooth rational curves E such that E·E=−1, L·E=2. The appendix contains an inductive argument in order to extend the result in higher dimension.     

On Manifolds Whose Tangent Bundle is Big and 1-Ample

A line bundle over a complex projective variety is called bigand 1-ample if a large multiple of it is generated by globalsections and a morphism induced by the evaluation of the spanningsections is generically finite and has at most 1-dimensionalfibers. A vector bundle is called big and 1-ample if the relativehyperplane line bundle over its projectivisation is big and1-ample. The main theorem of the present paper asserts that any complexprojective manifold of dimension 4 or more, whose tangent bundleis big and 1-ample, is equal either to a projective space orto a smooth quadric. Since big and 1-ample bundles are ‘almost’ample, the present result is yet another extension of the celebratedMori paper ‘Projective manifolds with ample tangent bundles’(Ann. of Math. 110 (1979) 593–606). The proof of the theorem applies results about contractionsof complex symplectic manifolds and of manifolds whose tangentbundles are numerically effective. In the appendix we re-provethese results. 2000 Mathematics Subject Classification 14E30,14J40, 14J45, 14J50.     

A remark on relative geometric invariant theory for quasi‐projective varieties

Relative geometric invariant theory studies the behavior of semistable points under equivariant morphisms. More precisely, suppose G is a reductive linear algebraic group over an algebraically closed field k, X and Y are quasi‐projective varieties endowed with G‐actions, is a G‐equivariant projective morphism, the G‐action on Y is linearized in the ample line bundle M, and the G‐action on X is linearized in the φ‐ample line bundle L. For any positive integer n, there is an induced linearization of the G‐action on X in the line bundle . If Y is projective and , the set of points in X that are semistable with respect to this linearization is contained in the preimage under φ of the set of points in Y that are semistable with respect to the given linearization in M. The same statement is trivially also true, if Y is affine and . In this note, we show by means of an example that the statement does not hold for arbitrary quasi‐projective varieties Y. This shows that a claim by Hu of the contrary is not true. Relative geometric invariant theory plays a role in the construction and study of degenerations of moduli spaces.     

Seshadri-exceptional foliations

 The maximal Seshadri number μ(L) of an ample line bundle L on a smooth projective variety X measures the local positivity of the line bundle L at a general point of X. By refining the method of Ein-Küchle-Lazarsfeld, lower bounds on μ(L) are obtained in terms of L ⁿ , n=dim(X), for a class of varieties. The main idea is to show that if a certain lower bound is violated, there exists a non-trivial foliation on the variety whose leaves are covered by special curves. In a number of examples, one can show that such foliations must be trivial and obtain lower bounds for μ(L). The examples include the hyperplane line bundle on a smooth surface in ℙ³ and ample line bundles on smooth threefolds of Picard number 1. Received: 29 June 2001 / Published online: 16 October 2002 RID="⋆" ID="⋆" Supported by Grant No. 98-0701-01-5-L from the KOSEF. RID="⋆⋆" ID="⋆⋆" Supported by Grant No. KRF-2001-041-D00025 from the KRF.     

Some remarks on ample line bundles on abelian varieties

Let L be an ample line bundle on an abelian variety A. We show that L² is very ample if (A,L) is not isomorphic to (A₁×A₂,o(D₁×A₂+A₁×D₂)) where A_i is an abelian variety (i=1,2), D_i is an ample divisor on A_i (i=1,2) and (A₁,o(D₁))=1, and if (A,L)2. As an application we show that L² is base point free if L is an ample line bundle on bielliptic surface.In conclusion, the author would like to thank the referee for very helpful advice.     

When K+(n-4)L fails to be nef

Let X be a smooth complex projective variety of dimension n and let L be an ample line bundle on X. We study polarized pairs (X,L) for which K+(n−3)L is nef but K+(n−4)L fails to be nef. Supported by MURST funds     

On the bicanonical map of a surface section of a threefold

Let (ℳ, ℒ) be a 3-fold of log-general type polarized by a very ample line bundle ℒ. We study the pairs (ℳ, ℒ) in the case when there exists at least one smooth surface Ŝ ∈ |ℒ| such that the bicanonical map associated to |2K_Ŝ| is not birational. As one consequence of our classification we obtain the result:if a smooth projective threefold has non- negative Kodaira dimension, then given any smooth very ample divisor Ŝon the threefold, the bicanonical map associated to |2K_Ŝ|is birational.     

Simultaneous generation of jets on K3 surfaces

Let L be an ample line bundle on a K3 surface. We give a sharp bound on n for which nL is k-jet ample.Received: 27 December 2002