Non-degenerate divisors on an algebraic surface
1960; Hiroshima University - Department of Mathematics; Volume: 24; Issue: 1 Linguagem: Inglês
10.32917/hmj/1555639738
ISSN0018-2079
Autores Tópico(s)Advanced Topics in Algebra
ResumoI.Introduction.The notion of non-degenerate divisors on an abelian variety was introduced first by Morikawa [2]Il, then after Weil proved that if Xis a non-digenerate divisor on an abelian variety, then the complete linear system / mX / is ample 2 l for a sufficiently large m [6].The latter property of divisors can be transferred to any variety and we shall call, in this paper, a divisor X on an algebraic variety "non-degenerate", if I mX I is ample and has no fixed component for a sufficiently large m.Now we ask how one can distinguish the class of non-degenerate divisors among the set of divisors.For this purpose we shall introduce the notion of (p)-cycles on a non-singular variety vn.Let X be an r-dimensional cycle on V.If the Kronecker index (X.Y) is positive for all positive cycles Y of codimension r, we shall call X a (p)-cycle.The main theorem in this paper asserts that if Vis a non-singular surface, any (p)-divisor is non-degenerate and conversely.On the other hand a nondegenerate divisor is a (p)-divisor and it is seen easily that any positive (p)divisor is non-degenerate on an abelian variety 3 l_ It would be interesting to investigate the relation between these two notions in general situation. Summary of known results. (A)Let F be a complete non-singular surface and let X be a divisor on F. Let2(X) be the sheaf of germs of rational
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