# zbMATH — the first resource for mathematics

Some applications of the theory of positive vector bundles. (English) Zbl 0547.14009
Complete intersections, Lect. 1st Sess. C.I.M.E., Acireale/Italy 1983, Lect. Notes Math. 1092, 29-61 (1984).
[For the entire collection see Zbl 0539.00006.]
The main purpose of these notes is to apply the theory of ample vector bundles to several concrete geometric problems. For instance we give a simple proof of a recent theorem of Ghione concerning the existence of special divisors associated to a vector bundle on an algebraic curve. We also show that if X is a smooth irreducible projective variety of dimension n, and if $$f:X\to {\mathbb{P}}^ n$$ is a branched covering of degree d, then the induced maps $$f_*:\pi_ i(X)\to\pi_ i({\mathbb{P}}^ n)$$ are bijective if $$i\leq n+1-d$$ and surjective if $$i=n+2-d$$. This is an analogue of the Barth-Larsen theorems for subvarieties of projective space. Finally we show how results of Mori lead to the proof of a conjecture of Remmert and Van de Ven to the effect that if X is a smooth projective variety of positive dimension which is the target of a surjective map $$f:{\mathbb{P}}^ n\to X,$$ then X is isomorphic to $${\mathbb{P}}^ n$$. - The notes also contain an elementary survey of the general theory of ample vector bundles.
We have found two potentially confusing misprints. First, in (1.1) the sheaf $${\mathcal F}$$ must be coherent. Secondly, in the definition of $$\rho^ r_ d(C,M)$$ in §2, the factor $$(g-d+1)$$ should have been (g- d-1).

##### MSC:
 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14F35 Homotopy theory and fundamental groups in algebraic geometry 14M12 Determinantal varieties