On the influence of the Segre conjecture on the Mori cone of blown-up surfaces

Details On the influence of the Segre conjecture on the Mori cone of blown-up surfaces On the influence of the Segre conjecture on the Mori cone of blown-up surfaces

2011-10-04

arxiv.org/abs/1110.0722v2

Mathematics

Algebraic Geometry

20 pages, 2 figures, Section 3 revisited

Scientific paper

In this paper we recall some important conjectures about linear systems of planar curves with given multiplicities at $r$ general points and we generalize these conjectures stating the Segre Conjecture for a smooth projective surface $Y$. We produce some counterexamples of Segre Conjecture and we translate these conjectures in terms of the Mori cone $\bar{NE}(X)$ of the blow up $X$ of $Y$ at general points $x_1, ..., x_r$. We generalize some known results and we show that if the Segre Conjecture holds true, then a part of the Mori cone $\bar{NE}(X)$ is circular and in fact it does coincide with a part of the positive cone of the blown-up surface $X$.

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