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

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