Algebraic Geometry

The paper generalizes some of the well-known results for K3 surfaces to higher-dimensional irreducible symplectic (or, equivalently, compact irreducible hyperkaehler) manifolds. In particular, we discuss the projectivity of such manifolds and their ample resp. Kaehler cones. It is proved that the period map surjects any non-empty connected component of the moduli space of marked manifolds onto the corresponding period domain. We also establish an anlogue of the transitivity of the Weyl-action on the set of chambers of a K3 surface. As a converse of a higher-dimensional version of the `Main Lemma' of Burns and Rapoport we prove that two birational irreducible symplectic manifolds are deformation equivalent. This compares nicely with a result of Batyrev and Kontsevich.

Let J be the jacobian of a reduced projective curve C with nodes only. 1) We give a simple and natural definition for its many compactifications and show the connection with various other definitions appearing in the literature. 2) Among all compactifications we choose one canonical, and define a theta divisor on it. 3) We give two very explicit and simple descriptions of a stratification of this canonical compactification into homogeneous spaces over J.

We construct natural relative compactifications for the relative Jacobian over a family X/S of reduced curves. In contrast with all the available compactifications so far, ours admit a universal sheaf, after an etale base change. Our method consists of considering the functor F of relatively simple, torsion-free, rank 1 sheaves on X/S , and showing that certain open subsheaves of F have good properties. Strictly speaking, the functor F is only representable by an algebraic space, but we show that F is representable by a scheme after an etale base change. Finally, we use theta functions originating from vector bundles to compare our new compactifications with the available ones.

We show that the algebraic and the symplectic GW-inivariants of smooth projective varieties are equivalent.

The main result of the paper is a boundedness for n -complements on algebraic surfaces. In addition, applications of this theorem to a classification of log Del Pezzo surfaces and of birational contractions for 3-folds are formulated.

Throughout this abstruct A will denote a noetherian commutative ring of dimension n . The paper has two parts. Among the interesting results in Part-1 are the following: 1) {\it suppose that f 1 , f 2 ,..., f r (with r≤n ) is a regular sequence in A and suppose Q is a projective A -module of rank r that maps onto the ideal ( f 1 , f 2 ,..., f r−1 , f (r−1)! r ) . Then [Q]=[ Q 0 ⊕A] in K 0 (A) for some projective A−module Q 0 of rank r−1 .} 2) The set F 0 K 0 (A)={[A/I]∈ K 0 (A):I is a locally complete intersection ideal in A of height n} is a {\it subgroup} of K 0 (A) . We also show that if A is a reduced affine algebra over a field k then F 0 K 0 (A) {\it is indeed the Zero Cycle Subgroup of} K 0 (A) {\it that is generated by smooth maximal ideals} $\Cal M$ {\it of height} n . 3){\it let A be such that whenever I is a locally complete intersection ideal of height n with [A/I]=0 then I is the image of a projective A−module of rank n . Then for any locally complete intersection ideal J of height n with [A/J] divisible by (n−1)! in F 0 K 0 (A) , there is a projective A−module of rank n that maps onto J }. The main result in Part-2 is the following construction: 1) {\it let X=SpecA be a Cohen-Macaulay scheme of dimension n and let r 0 , r be two integers with n/2≤ r 0 ≤r≤n . Let i) Q 0 be a projective A−module of rank r 0 −1 such that the restriction Q 0 |Y is trivial for all locally complete intersection subvarieties Y of codimension at least r 0 . Also ii) for k= r 0 to r , let I k be locally complete intersection ideals of height k so that I k / I 2 k has a generators of the type f 1 ,..., f k−1 , f (k−1)! k . Then there is a projective A−module Q of rank r such that

Two cycles on a projective variety over an algebraically closed field are shown to be rationally equivalent if and only if their difference equals a difference of complete intersections of a certain kind. Some of Bloch's conjectures for zero-cycles on surfaces can be restated more geometrically, which leads to several questions.

The paper contains a general construction which produces new examples of non simply-connected smooth projective surfaces. We analyze the resulting surfaces and their fundamental groups. Many of these fundamental groups are expected to be non-residually finite. Using the construction we also suggest a series of potential counterexamples to the Shafarevich conjecture which claims that the universal covering of smooth projective variety is holomorphically convex. The examples are only potential since they depend on group theoretic questions, which we formulate, but we do not know how to answer. At the end we formulate an arithmetic version of the Shafarevich conjecture.

Recently, there has been great interest in the application of composition laws to problems in enumerative geometry. Using the moduli space of stable maps, we compute the number of irreducible, reduced, nodal, degree- d genus- 2 plane curves whose normalization has a fixed complex structure and which pass through 3d−2 general points in P 2 .

Motivated by questions occuring in the construction of certain twistor spaces the parameter space of conics tangent to a given quartic is investigated. For a given real quartic surface in complex $\PP ^3$ that has exactly 13 ordinary nodes as singularities this parameter space is studied. In particular, its irreducible components are described.