On the degree of the discriminant locus of a smooth sectional surface of a (n+2)-fold with nonnegative Kodaira dimension

LetX be a smooth sectional surface of an (n+2)-fold with nonnegative Kodaira dimension. In this paper we improve Lanteri and Sommese estimates of the degree of the discriminant locus ofX whenn2.     

On moduli of vector bundles on surfaces with negative Kodaira dimension

Summary Here we prove the following result. Fix integersq, τ,a’, b’, a’ _i, 1≤i≤τ,a’, b’, a’ _i, 1≤i≤τ; then there is an integerew such that for every integert≥w, for every algebraically closed fieldK for every smooth complete surfaceX with negative Kodaira dimension, irregularityq andK _X ² =8(1−q)−τ, the following condition holds; ifX→S is a sequence fo τ blowing-downs which gives a relatively minimal model with ruling ρ:S→C, take as basis of the Neron Severi groupNS(X) a smooth rational curve which is the total transform of a fiber ofC, the total transform of a minimal section of ρ and the total transformD _i, 1≤i≤τ, of the exceptional curver; then for everyH andL∈Pic (X) withH ample,H (resp.L) represented by the integersa’, b’, a’ _i, (resp.a’, b’, a’ _i), 1≤i≤τ, in the chosen basis ofNS(X) the moduli spaceM(ZX, 2,H, L, t) of rank 2H-stable vector bundles onX with determinantL andc ₂=t is generically smooth and the number, dimension and ?birational structure? of the irreducible components ofM(X, 2,H, L, t)_red do not depend on the choice ofK andX. Furthermore the birational structure of these irreducible components can be loosely described in terms of the birational structure of the components of suitableM(S, 2,H’, L’, t’)_red withS a relatively minimal model ofX.

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Sunto SiaX una superficie algebrica liscia completa con dimensione di Kodaira negativa e definita su un campo algebricamente chiusoK; fissiamoH eL∈Pic (X),t∈Z; siaq l’irregolarità diX e τ≔8(1−q)−K _X ^Emphasis>2 ; siaM(X, 2,H, L, t) to schema dei moduli dei fibrati vettorialiH-stabili di rango 2 suX con determinateL ec ₂=t. Si dimostra che esiste una costantew che dipende solo daq, da τ e dalla classe numerica diH e diL (ma non da char (K) o dalla classe di isomorphismo diX) tale che per ognit≥w il numero, la dimensione e ?la struttura birazionale? delle componenti irriducibili diM(X, 2,H, L, t)_red non dipende dalla scelta di char (K),K eX ma solo daq, τ e dalle classi diH eL inNS(X). Inoltre la ?struttura birazionale? di queste componenti irriducibili può essere grossolanamente descritta in termini delle componenti di opportuni spazi di moduliM(S, 2,H’, L’, t’) (doveS è un modello minimale diX).

     

On the iteration of the adjunction process for surfaces of negative Kodaira dimension

Let (X,L) be a pair consisting of a smooth, complex, projective surface X and L a very ample line bundle on it. Suppose that the Kodaira dimension K(X) of X is negative. Then using the results obtained by A.J.Sommese and A.Van de Ven in [11], we find that the sectional genus, g_k=g(L_k), of the successive iterated minimal reductions (X_k,L_k), see 1. for the definition, reach a maximum value and then monotonically decrease to a final value g_n=g(L_n)g=g(L). This result gives a concrete way to express any pair (X,L), with K(X)=–, in terms of a minimal model.     

On the Kodaira dimension of minimal threefolds

Dedicated to Professor M. Nagata on his sixtieth birthday     

On a hyperplane section theorem of Gurjar

A hyperplane section theorem by R. V. Gurjar [3] is given a short proof. Power series in any number (at least three) of variables satsifying the condition of the theorem are explicitly constructed. In the course of the proof, the restrictive-looking condition of the theorem is given an easy sufficient condition from the view point of the weighted projective spaces. Received April 12, 2000 / Accepted August 21, 2000 / Published online October 30, 2000     

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.     

On the pluricanonical maps of varieties of intermediate Kodaira dimension

In this paper we will prove a uniformity result for the Iitaka fibration $f:X\rightarrow Y$ , provided that the generic fiber has a good minimal model and the Prokhorov–Shokurov conjecture holds. In particular, the result holds if the variation of $f$ is zero or $\kappa (X)=\text{ dim}X-1$ .     

On the discriminant locus of a Lagrangian fibration

Let be an irreducible holomorphic symplectic manifold of dimension 2n fibred over . Matsushita proved that the generic fibre is a holomorphic Lagrangian abelian variety. In this article we study the discriminant locus parametrizing singular fibres. Our main result is a formula for the degree of Δ, leading to bounds on the degree when X is a fourfold.     

On the number of minimal models of a log smooth threefold

We give a topological bound on the number of minimal models of a class of three-dimensional log smooth pairs of log general type.     

Wild automorphisms of varieties with Kodaira dimension 0

An automorphism σ of a projective variety X is said to be wild if σ(Y) ≠ Y for every non-empty subvariety Y \subsetneq X{Y \subsetneq X} . In [1] Z. Reichstein, D. Rogalski, and J.J. Zhang conjectured that if X is an irreducible projective variety admitting a wild automorphism then X is an abelian variety, and proved this conjecture for dim(X) ≤ 2. As a step toward answering this conjecture in higher dimensions we prove a structure theorem for projective varieties of Kodaira dimension 0 admitting wild automorphisms. This essentially reduces the Kodaira dimension 0 case to a study of Calabi-Yau varieties, which we also investigate. In support of this conjecture, we show that there are no wild automorphisms of certain Calabi-Yau varieties.     

On the genus of a hyperplane section of a geometrically ruled surface

Summary In this paper we estimate the minimal genus of hyperplane sections of a geometrically ruled surface.     

The general hyperplane section of a curve

In this paper, we discuss some necessary and sufficient conditions for a curve to be arithmetically Cohen-Macaulay, in terms of its general hyperplane section. We obtain a characterization of the degree matrices that can occur for points in the plane that are the general plane section of a non-arithmetically Cohen-Macaulay curve of . We prove that almost all the degree matrices with positive subdiagonal that occur for the general plane section of a non-arithmetically Cohen-Macaulay curve of , arise also as degree matrices of some smooth, integral, non-arithmetically Cohen-Macaulay curve, and we characterize the exceptions. We give a necessary condition on the graded Betti numbers of the general plane section of an arithmetically Buchsbaum (non-arithmetically Cohen-Macaulay) curve in . For curves in , we show that any set of Betti numbers that satisfies that condition can be realized as the Betti numbers of the general plane section of an arithmetically Buchsbaum, non-arithmetically Cohen-Macaulay curve. We also show that the matrices that arise as a degree matrix of the general plane section of an arithmetically Buchsbaum, integral, (smooth) non-arithmetically Cohen-Macaulay space curve are exactly those that arise as a degree matrix of the general plane section of an arithmetically Buchsbaum, non-arithmetically Cohen-Macaulay space curve and have positive subdiagonal. We also prove some bounds on the dimension of the deficiency module of an arithmetically Buchsbaum space curve in terms of the degree matrix of the general plane section of the curve, and we prove that they are sharp.

     

Numerical analogues of the Kodaira dimension and the Abundance Conjecture

We add further notions to Lehmann’s list of numerical analogues to the Kodaira dimension of pseudo-effective divisors on smooth complex projective varieties, and show new relations between them. Then we use these notions and relations to fill in a gap in Lehmann’s arguments, thus proving that most of these notions are equal. Finally, we show that the Abundance Conjecture, as formulated in the context of the Minimal Model Program, and the Generalized Abundance Conjecture using these numerical analogues to the Kodaira dimension, are equivalent for non-uniruled complex projective varieties.     

The Kodaira dimension of the moduli of K3 surfaces

The global Torelli theorem for projective K3 surfaces was first proved by Piatetskii-Shapiro and Shafarevich 35 years ago, opening the way to treating moduli problems for K3 surfaces. The moduli space of polarised K3 surfaces of degree 2d is a quasi-projective variety of dimension 19. For general d very little has been known hitherto about the Kodaira dimension of these varieties. In this paper we present an almost complete solution to this problem. Our main result says that this moduli space is of general type for d>61 and for d=46, 50, 54, 57, 58, 60.     

A note on threefolds with a hyperplane section of general type

This paper deals with polarized pairs , where is a nonsingular projective threefold and is a very ample line bundle on it, such that for one smooth member  | |, one has (Â)=2. A large class of pairs whose adjoint line bundle is nef and big was indirectedly studied by Beltrametti and co-workers. We add some more information, both in this general case and also when the adjoint line bundle fails to be nef and big.     

On the critical dimension of a fourth order elliptic problem with negative exponent

We study the regularity of the extremal solution of the semilinear biharmonic equation on a ball B⊂RN, under Navier boundary conditions u=Δu=0 on ∂B, where λ>0 is a parameter, while τ>0, β>0 are fixed constants. It is known that there exists λ^∗ such that for λ>λ^∗ there is no solution while for λ<λ^∗ there is a branch of minimal solutions. Our main result asserts that the extremal solution u^∗ is regular (supBu^∗<1) for N?8 and β,τ>0 and it is singular (supBu^∗=1) for N?9, β>0, and τ>0 with small. Our proof for the singularity of extremal solutions in dimensions N?9 is based on certain improved Hardy-Rellich inequalities.