IJPAM: Volume 15, No. 1 (2004)

NODAL PROJECTIVE CURVES (MAINLY IN
THE PLANE) AND GEOMETRIC -NORMALITY

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

Abstract.Let $C \subset {\bf {P}}^r$ be an integral curve and $f: X \to C$ its normalization. Set $A:= T{\bf {P}}^r(-1)$ and $V:= H^0({\bf {P}}^r,\mathcal {O}_{{\bf {P}}^r}(1)) \cong H^0({\bf {P}}^r,A)^\ast$ . is said to be geometrically -normal (resp. rank geometrically -normal), , if the map $V^{\otimes k} \to H^0(X,f^\ast (\mathcal {O}_C(k)))$ (resp. $(V^\ast )^{\otimes k} \to H^0(X,f^\ast (A ^{\otimes k}))$ ) is surjective. Here we prove the geometric -normality and geometric rank two -normality for general nodal plane curves (with not too nodes) and certain nodal curves in ${\bf {P}}^r$ , $r \ge 3$ .