bounded by the other. Serre showed that it is possible for h1,0 to vanish while h0,1 is positive, while Mumford showed that for Enriques surfaces in characteristic 2 it is possible for h0,1 to vanish while h1,0 is positive. Alexander Grothendieck gave a complete description of the relation of q to in all characteristics. The dimension of the tangent space to the Picard scheme (at any point) is equal to . In characteristic 0 a result of Pierre Cartier showed that all groups schemes of finite type are non-singular, so the dimension of their tangent space is their dimension. On the other hand, in positive characteristic it is possible for a group scheme to be non-reduced at every point so that the dimension is less than the dimension of any tangent space, which is what happens in Igusa's example. Mumford shows that the tangent space to the Picard variety is the subspace of H0,1 annihilated by all Bockstein operations from H0,1 to H0,2, so the irregularity q is equal to h0,1 if and only if all these Bockstein operations vanish. References Algebraic surfaces Complex surfaces