The geometric approach to hydrodynamics was developed by Arnold to study Lagrangian stability of ideal fluids. It identifies a Lagrangian fluid flow with a geodesic on the Riemannian manifold of volume-preserving diffeomorphisms. The curvature of this manifold is typically negative but sometimes positive, and positivity leads to conjugate points (where initially close geodesics spread apart and come together again). In this talk we suppose a fluid in 3 satisfies a pointwise version of the Beale-Kato-Majda criterion for blowup at a finite time T. I will describe a theorem which states that either the geodesic experiences an infinite sequence of consecutive conjugate pairs approaching the blowup time, or the deformation tensor has a fairly special form at the blowup time. The first possibility suggests that one could "see" blowup geometrically in a weak space, such as the space of L2 measure-preserving transformations.