On the sectional geometric genus of quasi-polarized varieties,II

Let (X,L) be a quasi-polarized variety of dim X=n. In the previous paper we gave a new invariant (the i-th sectional geometric genus) of (X,L), which is a generalization of the degree and the sectional genus of (X,L). In this paper we study some properties of the sectional geometric genus, and we consider the i-th sectional geometric genus of some special varieties. This research was partially supported by Grant-in-Aid for Scientific Research (No.14740018) from the Ministry of Education, Culture, Sports, Science and Technology.Mathematics Subject Classification (2000):Primary 14C20; Secondary 14J30, 14J32, 14J35, 14J40, 14J45, 14N99     

On Fano 3-folds

In this paper Fano 3-folds of the principal series and first species are classified.Partially supported by NSF.     

On toroidal embeddings of 3-folds

Using combinatorial methods, we classify all birational morphisms blowingm ≦ 5 divisors down to a point. Those which do not factor through the blowing up of a point are treated in the body of the paper, and the factorizable morphisms are computed in an appendix. The relevance of this classification to the determination of the nature of the “general” toric morphism is discussed.     

On Fano indices of -Fano 3-folds

We shall give the best possible upper bound of the Fano indices together with a characterization of those -Fano 3-folds which attain the maximum in terms of graded rings.     

On anti-pluricanonical systems of Q-Fano 3-folds

We investigate birationality of the anti-pluricanonical map Φ-m,the rational map defined by the anti-pluricanonical system |-mKX|,on weak Q-Fano 3-folds.     

On the dimension of global sections of adjoint bundles for polarized 3-folds and 4-folds

Let (X,L) be a polarized manifold of dimension n defined over the field of complex numbers. In this paper, we treat the case where n=3 and 4. First we study the case of n=3 and we give an explicit lower bound for h⁰(K_X+L) if κ(X)≥0. Moreover, we show the following: if κ(K_X+L)≥0, then h⁰(K_X+L)>0 unless κ(X)=−∞ and h¹(O_X)=0. This gives us a partial answer of Effective Non-vanishing Conjecture for polarized 3-folds. Next for n=4 we investigate the dimension of H⁰(K_X+mL) for m≥2. If n=4 and κ(X)≥0, then a lower bound for h⁰(K_X+mL) is obtained. We also consider a conjecture of Beltrametti-Sommese for 4-folds and we can prove that this conjecture is true unless κ(X)=−∞ and h¹(O_X)=0. Furthermore we prove the following: if (X,L) is a polarized 4-fold with κ(X)≥0 and h¹(O_X)>0, then h⁰(K_X+L)>0.     

On the second sectional H-arithmetic genus of polarized manifolds

Let (X,L) be a polarized variety defined over the complex number field with dim X=n. In this paper we introduce the notion of the i-th sectional H-arithmetic genus ^H_i(X,L) for every integer i with 0in. We expect that this invariant has a property similar to the Euler-Poincaré characteristic of the structure sheaf of i-dimensional varieties. In this paper, we consider the case where X is smooth and i=2, and we study a polarized version of some results in the theory of surfaces.Mathematics Subject Classification (2000): 14C20, 14C17, 14C40, 14J29, 14J30, 14J35, 14J40This research was partially supported by the Grant-in-Aid for Young Scientists (B) (No.14740018), The Ministry of Education, Culture, Sports, Science and Technology, Japan.     

On the non existence of some rational 3-folds in ℙ5

In this paper we show that there are no smooth rational 3-folds in ?⁵ (C) which are rational conic bundles, over minimal surfaces, whose generic fibre is embedded as a rational curve of degreeh≥3, (ifh=2 there is a complete classification for these 3-folds as well as for the case of ?¹-bundles). Except for conic bundles, we also give the complete list of rational 3-folds in ?⁵ which are minimal according to Mori’s theory. These are little steps towards the classification of all smooth 3-folds in ?⁵ not of general type.     

Note on Fano 3-folds

LetE be an ample vector bundle of rankr on a compact complex manifoldX of dimension 3 with detE=–K _x, andi(X) the index ofX. Then it is proved in this note thati(X)r unless (X,E)(¹ × ²,p*O(1) q*), wherep,q are the projections and is isomorphic toO(2) O(1) or the tangent bundleT of ². This result gives a counterexample to the conjecture formed by T. Peternell.     

The sectional genus of quasi-polarised varieties

T. Fujita conjectured that the sectional genus of a quasi-polarised variety is non-negative. We prove this conjecture. Using the minimal model program we also prove that if the sectional genus is zero the Δ-genus is also zero. This leads to a birational classification of quasi-polarised varieties with sectional genus zero.     

On the classification of toric Fano 4-folds

The biregular classification of smoothd-dimensional toric Fano varieties is equivalent to the classification of special simplicial polyhedraP in ℝ ^d , the so-called Fano polyhedra, up to an isomorphism of the standard lattice . In this paper, we explain the complete biregular classification of all 4-dimensional smooth toric Fano varieties. The main result states that there exist exactly 123 different types of toric Fano 4-folds somorphism. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory Vol. 56. Algebraic Geometry-9, 1998.     

2-Spanned surfaces of sectional genus six

Sunto Un fibrato lineare olomorfo su una superficie algebrica proiettiva complessa non singolare S è detto k-spanned se le sue sezioni separano opportuni 0-cicli di S di lunghezza k + 1 — da (k + 1)-uple di punti distinti sino a getti di ordine k — e forniscono pertanto una immersione proiettiva di S di ordine superiore. Si fornisce un contributo alla classificazione proiettiva delle superfici determinando quelle di genere sezionale 6 immerse da un fibrato 2-spanned e quelle di genere sezionale 7 immerse da un fibrato 3-spanned.     

Vector bundles on ruled Fano 3-folds

Let be a ruled Fano 3-fold. The goal of this paper is to compute the dimension, prove the irreducibility and smoothness and describe the structure of the moduli space M _L (2;c ₁,c ₂) of L-stable, rank 2 vector bundles E on X with certain Chern classes and for a suitable polarization L closely related to c ₂. More precisely, we will cover the study of some moduli spaces M _L (2;c ₁,c ₂) such that the generic point is given as a non-trivial extension of line bundles. This work nicely reflects the general philosophy that moduli spaces inherits a lot of geometrical properties of the underlying variety. Received: 16 February 1999 / Revised version: 2 July 1999