Abstract:
We define generalizations of the Albanese variety for a projective variety X. The generalized Albanese morphisms X --> Alb_r(X) contract those curves C in X for which the induced morphism Hom(\pi_1(X),U(r)) --> Hom(\pi_1(C),U(r)) has a finite image. Thus, they may be interpreted as a U(r)-version of the Shafarevich morphism.

Abstract:
We construct vector bundles $R^r_\mu$ on a smooth projective curve $X$ having the property that for all sheaves $E$ of slope $\mu$ and rank $r$ on $X$ we have an equivalence: $E$ is a semistable vector bundle $\iff$ $Hom(R^r_\mu,E)=0$. As a byproduct of our construction we obtain effective bounds on $r$ such that the linear system $|R \cdot \Theta|$ has base points on the moduli space $U_X(r,r(g-1))$.

Abstract:
We study sheaves E on a smooth projective curve X which are minimal with respect to the property that $h^0(E \otimes L) >0$ for all line bundles L of degree zero. We show that these sheaves define ample divisors D(E) on the Picard torus Pic(X). Next we classify all minimal sheaves of rank one and two. As an application we show that the moduli space parameterizing rank two bundles of odd degree can be obtained as a Quot scheme.

Abstract:
We develop a formula (Theorem 5.1) which allows to compute top Chern classes of vector bundles on the vanishing locus $V(s)$ of a section of this bundle. This formula particularly applies in the case when $V(s)$ is the union of locally complete intersections giving the individual contribution of each component and their mutual intersections. We conclude with applications to the enumeration of rational curves in complete intersections in projective space.

Abstract:
We study base points of the generalized Theta-divisor on the moduli space of vector bundles on a smooth algebraic curve X of genus g defined over an algebraically closed field. To do so, we use the derived categories D(Pic(X)), D(Jac(X)), and the equivalence between them given by the Fourier-Mukai transform coming from the Poincar\'e bundle. The vector bundles P(m) on the curve X defined by Raynaud play a central role in this description. Indeed, we show that a vector bundle E is a base point of the generalized Theta-divisor, if and only if there exists a nontrivial homomorphism P(rk(E)g+1) --> E.

Abstract:
We show for the moduli space of rank-2 coherent sheaves on an algebraic surface that there exists a 'dual' moduli space. This dual space allows a construction of the first one without using the GIT construction. Furthermore, we obtain a Barth-morphism, generalizing the concept of jumping lines. This morphism is by construction a finite morphism.

Abstract:
Let $X$ be a smooth variety defined over an algebraically closed field of arbitrary characteristic and $\O_X(H)$ be a very ample line bundle on $X$. We show that for a semistable $X$-bundle $E$ of rank two, there exists an integer $m$ depending only on $\Delta(E).H^{\dim(X)-2}$ and $H^{\dim(X)}$ such that the restriction of $E$ to a general divisor in $|mH|$ is again semistable. As corollaries we obtain boundedness results, and weak versions of Bogomolov's theorem and Kodaira's vanishing theorem for surfaces in arbitrary characteristic.

Abstract:
Let X be an irreducible smooth projective curve defined over complex numbers, S= {p_1, p_2,...,p_n} \subset X$ a finite set of closed points and N > 1 a fixed integer. For any pair (r,d) in Z X Z/N, there exists a parabolic vector bundle R_{r,d,*} on X, with parabolic structure over S and all parabolic weights in Z/N, that has the following property: Take any parabolic vector bundle E_* of rank r on X whose parabolic points are contained in S, all the parabolic weights are in Z/N and the parabolic degree is d. Then E_* is parabolic semistable if and only if there is no nonzero parabolic homomorphism from R_{r,d,*} to E_*.

Abstract:
It is known that a vector bundle E on a smooth projective curve Y defined over an algebraically closed field is semistable if and only if there is a vector bundle F on Y such that the cohomologies of E\otimes F vanish. We extend this criterion for semistability to vector bundles on curves defined over perfect fields. Let X be a geometrically irreducible smooth projective curve defined over a perfect field k, and let E be a vector bundle on X. We prove that E is semistable if and only if there is a vector bundle F on $X$ such that the cohomologies of E\otimes F vanish. We also give an explicit bound for the rank of $F$.