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).

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 |