Three dimensional canonical singularities in codimension two in positive characteristic

We investigate local structure of a three dimensional variety X defined over an algebraically closed field k of characteristic p > 0 with at most canonical singularities. Under the assumption that p ≥ 3 and a general hyperplane cut of X has at most rational singularities, we show that local structure of X in codimension two is well understood in the level of local equations. Consequently, we find that i) any singularity of such a variety X in codimension two is compound Du Val, ii) it has a crepant resolution, iii) it is analytically a product of a rational double point and a nonsingular curve when p ≥ 3 with two exceptions in p = 3, and iv) R ¹π*O _X~=R1π*KX~=0 holds outside some finite points of X for any resolution of singularities π:X~→X.