IJPAM: Volume 18, No. 3 (2005)

ON THE GAUSSIAN MAPS OF BLOWN-UP
PROJECTIVE MANIFOLDS

E. Ballico
Department of Mathematics
University of Trento
380 50 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: ballico@science.unitn.it

Abstract.Let be a smooth projective manifold, $L, M\in \mbox{\rm Pic}(S)$ and $\gamma _{L,M}: H^0(S,L)\otimes H^0(S,M) \to H^0(S,\Omega _S \otimes L\otimes M)$ the Gaussian (or Wahl) map defined by $\gamma _{L,M}(f\otimes h) = fd(h)-hd(f)$ . Here we study the surjectivity of $\gamma _{L,M}$ for suitable , when is the blowing up at finitely many points of a nice manifold (e.g. a projective space). We obtain the surjectivity of suitable Gaussian maps for several subvarieties of and in particular the surjectivity of $\gamma _{\omega _X,\omega _X}$ for several smooth curves .