Higher order secant varieties and blown up varieties

Let π:X→Y be the blowing up of the projective varietyY at s general points. Here we study the higher order secant varieties of the linearly normal embeddings ofX andY into projective spaces. We give conditions on the embedding ofY which imply that the firstt secant varieties of a related embedding ofX have the expected dimension.     

On the global sections of a vector bundle over a projective variety

Here we study a few general formal properties related to the dimensions of the spaces of global sections of subsheaves of a vector bundle.

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Sunto SiaE un fibrato vettoriale algebrico su una varietà proiettiva. In questo lavoro studiamo alcune nozioni correlate alle dimensioni degli spazi di sezioni globali dei sottofasci diE.

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The author was partially supported by MURST and GNSAGA of CNR (Italy).     

On The Postulation of The Multiples of The Nodes for Generic Nodal Curves on Generic Rational Surfaces

Fix integers x > 0, m₁ ≥ … ≥ ^m _x > 0 and P₁,…,P_x ∈ P² such that no 3 of them are collinear. Let C ? P² a “ general ” degree d plane curve with an ordinary point with multiplicity m _i at each P _i and y further singularities which are ordinary nodes. Fix any A ? Sing(C){P₁,…, P_x} and any integer m > 0. Here we study the postulation of the fat points m A ? ∪_Q∈AmQ.     

Exact Sequences of Semistable Vector Bundles on Algebraic Curves

Let X be a smooth complex projective curve of genus g 1. Ifg 2, then assume further that X is either bielliptic or withgeneral moduli. Fix integers r, s, a, b with r > 1, s >1 and as br. Here we prove the existence of an exact sequence [formula] of semistable vector bundles on X with rk(H) = r, rk(Q) = s,deg(H) = a and deg(Q) = b. 1991 Mathematics Subject Classification14H60.     

Generic curves of small genus inP 3 are of maximal rank

Mathematische Annalen -     

Uniform vector bundles on quadrics

Summary In this paper we prove that a rankr uniform vector bundle on a nonsingular quadricQ with dimQ≥2r+2 is a direct sum of line bundles. We study also rank 2 uniform vector bundles onP ₁×P₁. We prove that they are not all homogeneous and that any rank 2 homogeneous vector bundles onP ₁×P₁ is decomposable.

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Riassunto In questo lavoro si dimostra che un fibrato uniforme di rangor su una quadrica non singolareQ con dimQ≥2r+2 è somma diretta di fibrati in rette. Si studiano poi i fibrati uniformi di rango 2 suP ₁×P₁. Si dimostra che non sono tutti omogenei e che ogni fibrato omogeneo di rango 2 suP ₁×P₁ è decomponibile.

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The author is member of G.N.S.A.G.A. of C.N.R.