On the effective cone of $\mathbb{P}^n$ blown-up at $n+3$ points

Maria Chiara Brambilla; Olivia Dumitrescu; Elisa Postinghel

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arXiv · 1501.04094

On the effective cone of $\mathbb{P}^n$ blown-up at $n+3$ points

Maria Chiara Brambilla·Olivia Dumitrescu·Elisa Postinghel

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Abstract

We compute the facets of the effective and movable cones of divisors on the blow-up of $\mathbb{P}^n$ at $n+3$ points in general position. Given any linear system of hypersurfaces of $\mathbb{P}^n$ based at $n+3$ multiple points in general position, we prove that the secant varieties to the rational normal curve of degree $n$ passing through the points, as well as their joins with linear subspaces spanned by some of the points, are cycles of the base locus and we compute their multiplicity. We conjecture that a linear system with $n+3$ points is linearly special only if it contains such subvarieties in the base locus and we give a new formula for the expected dimension.

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BibTeX RIS

Maria Chiara Brambilla, Olivia Dumitrescu, Elisa Postinghel. 2015-01-16. On the effective cone of $\mathbb{P}^n$ blown-up at $n+3$ points. https://arxiv.org/abs/1501.04094

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