Helly-type theorems for varieties (Q1124641)

Let \(\Gamma\) be a commutative field and let \(\Gamma [X_ 0,...,X_ n]\) be the ring of all homogeneous polynomials in \(n+1\) variables over \(\Gamma\). Main theorem: Let \(d\geq 1\) and let V be a d-dimensional vector subspace of \(\Gamma [X_ 0,...,X_ n]\). Then V has the (d-1)-Helly property. Moreover, if \(\Gamma\) is infinite, then V does not have the (d-2)-Helly property.