Consider a projective variety X subset of 1n (over an algebraically closed field of characteristic zero), together with a (reduced) simple normal crossings divisor E subset of 1n, where the degrees of both X and E are at most d. We show there is a pair (n ', d ') which can be explicitly computed in terms of (n, d), such that (X, E) has a log resolution of singularities (X ', E '), where (X ', E ') can be embedded in 1n ' and both X ' and E ' have degrees at most d ' in 1n '.
We bound the indices of normal abelian subgroups in finite groups contained in the Cremona group of rank 2 over a field of odd characteristic.
We prove the base point free theorem for log canonical foliated pairs of rank one on a Q-factorial projective klt threefold. Moreover, we show abundance in the case of numerically trivial log canonical foliated pairs of rank one in any dimension.
