Dedicated to the blessed memory of Andrei Nikolaevich Tyurin – adviser and friend

The variety of sums of powers of a homogeneous polynomial of degree d in n variables is defined and investigated in some examples, old and new. These varieties are studied via apolarity and syzygies. Classical results (cf. [Sylvester 1851], [Hilbert 1888], [Dixon, Stuart 1906]) and some more recent results of Mukai (cf. [Mukai 1992]) are presented together with new results for the cases (n, d) = (3, 8), (4, 2), (5, 3). In the last case the variety of sums of 8 powers of a general cubic form is a Fano 5-fold of index 1 and degree 660.

this paper was partially supported by the National Science Foundation.

this paper we prove the other direction. Specifically, we prove the following theorem.

A known result by Horrocks (see [H]) characterizes the line bundles on a projective space as the only indecomposable vector bundles without intermediate cohomology. This result has been generalized by Ottaviani (see [O1], [O2]) to quadrics and Grassmannians.

Abstract. In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological method using higher almost split sequences and results from birational geometry. We obtain a large class of 2-CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2-CY tilted algebras for simple/minimally elliptic curve singuralities.

Abstract. In this paper we show that on a general sextic hypersurface X ⊂ P 4, a rank 2 vector bundle E splits if and only if h 1 (E(n)) = 0 for any n ∈ Z. We get thus a characterization of complete intersection curves in X. 1.

Abstract. We give a partial positive answer to a conjecture of Tyurin ([28]). Indeed we prove that on a general quintic hypersurface of P 4 every arithmetically Cohen–Macaulay rank 2 vector bundle is infinitesimally rigid. 1.

Abstract. Let W be a finite dimensional representation of a linearly reductive group G over a field k. Motivated by their work on classical rings of invariants, Levasseur and Stafford asked whether the ring of invariants under G of the symmetric algebra of W has a simple ring of differential operators. In this paper, we show that this is true in prime characteristic. Indeed, if R is a graded subring of a polynomial ring over a perfect field of characteristic p> 0 and if the inclusion R ֒ → S splits, then Dk(R) is a simple ring. In the last section of the paper, we discuss how one might try to deduce the characteristic zero case from this result. As yet, however, this is a subtle problem and the answer to the question of Levasseur and Stafford remains open in characteristic

Abstract. Let X be a general complex Fano threefold of genus 8. We prove that the moduli space of rank two semistable sheaves on X with Chern numbers c1 = 1, c2 = 6 and c3 = 0 is isomorphic to the Fano surface F(X) of conics on X. This surface is smooth and isomorphic to the Fano surface of lines in the orthogonal to X cubic threefold. Inside F(X), the non-locally free sheaves are parameterized by a smooth curve of genus 26 isomorphic to the base of the family of lines on X. 1.