On the Osculating Bundles of Curves inPn

LetXbe a smooth projective curve of genusg,L Picd(X),V H0(X, L),VspanningL. For every integertwith 1 ≤ t ≤ n − 2 dim(V) − 3, letPt(L) be the principal bundle of ordertofLandGt(L) Pt(L) the image of the Taylor expansion mapV OX → Pt(L), i.e., Pieneʹs osculating bundle of ordertofL. If rank(Gt(L)) = t + 1,c deg(Gt(L)) − deg(Pt(L)) is the degree of the cuspidal locus of order ≤ tof the linear system V. Here we studyGt(L) as an abstract bundle onXusing elementary transformations. For everyg ≥ 2 we obtain conditions ong,d,n,t, andcsuch that for allXand allL Picd(X), there isV H0(X, L) such that the associated osculating bundleGt(L) is a general stable bundle of rankt + 1 and degree (t + 1)t(g − 1) + (t + 1)d − c.