Elena Rubei

Quivers and the cohomology of homogeneous vector bundles

We describe the cohomology groups of a homogeneous vector bundle E on any Hermitian symmetric variety X = G/P of ADE-type as the cohomology of a complex explicitly described. The main tool is the equivalence (introduced by Bondal, Kapranov, and Hille) between the category of homogeneous bundles and the category of representations of a certain quiver QX with relations. We prove that the relations are the commutative ones on projective spaces, but they involve additional scalars on general Grassmannians. In addition, we introduce moduli spaces of homogeneous

A result on resolutions of Veronese embeddings

SuntoL’argomento di questo articolo sono le sizigie degli ideali delle varietà di Veronese. Per il teorema di Green sappiamo cheOPn(d) soddisfa la proprietàNp di Green-Lazarsfeld ∀d≥p, ∀n. Per il teorema di Ottaviani-Paoletti sen≥2, d≥3 and 3d−2≤p alloraOPn(d) non soddisfa la ProprietàNp. I casin≥3, d≥3, d<p<3d−2 sono ancora aperti (eccetton=d=3). Qui consideriamo uno di tali casi, precisamente proviamo cheOPn(3) soddisfa la ProprietàN4 ∀n. Inoltre proviamo cheOPn(d) soddisfaNp ∀n≥p se e solo seOPp(d) satisfiesNp.AbstractThis paper deals with syzygies of the ideals of the Veronese embeddings. By Green’s Theorem we know thatOPn(d) satisfies Green-Lazarsfeld’s PropertyNp ∀d≥p, ∀n. By Ottaviani-Paoletti’s theorem ifn≥2, d≥3 and 3d−2≤p thenOPn(d) does not satisfy PropertyNp. The casesn≥3, d≥3, d<p<3d−2 are still open (exceptn=d=3). Here we deal with one of these cases, namely we prove thatOPn(3) satisfies PropertyN4 ∀n. Besides we prove thatOPn(d) satisfiesNp ∀n≥p iffOPn(d) satisfiesNp.

On syzygies of Segre embeddings

We study the syzygies of the ideals of the Segre embeddings. Let d ∈ N, d > 3; we prove that the line bundle O(1,..., 1) on the P 1 ×... x P 1 (d copies) satisfies Property Np of Green-Lazarsfeld if and only if p ≤ 3. Besides we prove that if we have a projective variety not satisfying Property Np for some p, then the product of it with any other projective variety does not satisfy Property Np. From this we also deduce other corollaries about syzygies of Segre embeddings.

On dissimilarity vectors of general weighted trees

Abstract Let T be a (not necessarily positive) weighted tree with n leaves numbered by the set { 1 , … , n } . For any i 1 , … , i k ∈ { 1 , … , n } , define D i 1 , … , i k ( T ) to be the sum of the lengths of the edges of the minimal subtree joining i 1 , … , i k . We will call such numbers “ k -weights” of the tree and we call the k -weights for any k ≥ 2 “multi-weights” of the tree. In this paper, we give a characterization of the families of real numbers that are the families of the multi-weights of a tree.

On syzygies of abelian varieties

In this paper we prove the following result: Let X be a co:mplex torus and M a normally generated line bundle on X; then, for every p > 0, the line bundle MP+1 satisfies Property Np of Green-Lazarsfeld.

Stability of homogeneous bundles on P 3

AbstractWe study the stability and the simplicity of some homogeneous bundles on P3 by using the quiver associated to homogeneous bundles introduced by Bondal and Kapranov in [3]. In particular we show that the homogeneous bundles on P3 whose quiver support is a parallelepiped or a classical staircase are stable. For instance the bundles E whose minimal free resolution is of the kind

Lazzeri's Jacobian of oriented compact Riemannian manifolds

0 \rightarrow S^{\lambda_1, \lambda_2, \lambda_3} V (t) \rightarrow S^{\lambda_1+s, \lambda_2, \lambda_3} V (t+s) \rightarrow E \rightarrow 0

Weighted Graphs with Distances in Given Ranges

are stable.