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

ABSTRACT

We compute the facets of the effective and movable cones of divisors on the blow-up of at n + 3 points in general position. Given any linear system of hypersurfaces of 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.

KEYWORDS:

• Fröberg–Iarrobino conjectures
• effective and movable cones
• rational normal curve
• secant varieties

2000 AMS Subject Classification:

• Primary 14C20
• Secondary 14J70
• 14J26
• 13D40

Funding

The authors would like to thank the Research Center FBK-CIRM Trento for the hospitality and financial support during the stay for the Summer School “An interdisciplinary approach to tensor decomposition” (Summer 2014) and during their one month “Research in Pairs” program (Winter 2015). The first author is partially supported by MIUR and INDAM. The second author is a member of the Simion Stoilow Institute of Mathematics of the Romanian Academy. The third author is supported by the Research Foundation – Flanders (FWO).