On the arithmetic rank of certain Segre products

We compute the arithmetic ranks of the defining ideals of homogeneous coordinate rings of certain Segre products arising from elliptic curves. The cohomological dimension of these ideals varies with the characteristic of the field, though the arithmetic rank does not. We also study the related set-theoretic Cohen-Macaulay property for these ideals. In [12] Lyubeznik writes: Part of what makes the problem about the number of defining equations so interesting is that it can be very easily stated, yet a solution, in those rare cases when it is known, usually is highly nontrivial and involves a fascinating interplay of Algebra and Geometry. In this note we present one of these rare cases where a solution is obtained: for a smooth elliptic curve E ⊂ P, we determine the arithmetic rank of the ideal a defining the Segre embedding E×P1 ⊂ P, and exhibit a natural generating set for a up to radical. The ideal a is not a set-theoretic complete intersection and, in the case of characteristic zero, we use reduction modulo p methods to prove moreover that a is not set-theoretically Cohen-Macaulay. 1. The Segre embedding of E × P Let A and B be N-graded rings over a field A0 = B0 = K. The Segre product of A and B is the ring