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On the arithmetic rank of projective subspace arrangements

Research paper by Stefan Tohaneanu

Indexed on: 07 Jul '16Published on: 07 Jul '16Published in: Mathematics - Algebraic Geometry



Abstract

If $V$ is a projective subspace arrangement over an infinite field, consisting of $m$ irreducible components all intersecting in the empty set, and each of codimension $c_i,i=1,\ldots,m$, then the arithmetic rank of the defining ideal of $V$ is bounded above by $\displaystyle 1+\sum_{i=1}^m(c_i-1)$. If the base field is algebraically closed, the bound is attainable by an entire class of examples that includes Hartshorne's example. The result comes as a corollary to an upper bound on the arithmetic rank of generalized star configuration varieties, and to a result that shows that any projective subspace arrangement can be interpolated by such a variety.