On the Effective Cone of Pn Blown-up at n + 3 Points

© 2016 Taylor & Francis.We compute the facets of the effective and movable cones of divisors on the blow-up of ℙn at n + 3 points in general position. Given any linear system of hypersurfaces of ℙn 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, aswell as their joinswith linear subspaces spanned by some of the points, are cycles of the base locus andwe compute their multiplicity.We conjecture that a linear systemwith 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.