The genus-minimizing property of algebraic curves

Kronheimer, P. B.

doi:10.1090/S0273-0979-1993-00399-9

The genus-minimizing property of algebraic curves
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by P. B. Kronheimer PDF

Bull. Amer. Math. Soc. 29 (1993), 63-69 Request permission

Abstract:

A viable and still unproved conjecture states that, if X is a smooth algebraic surface and C is a smooth algebraic curve in X, then C realizes the smallest possible genus amongst all smoothly embedded 2-manifolds in its homology class. A proof is announced here for this conjecture, for a large class of surfaces X, under the assumption that the normal bundle of C has positive degree.

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