We establish the minimal model program for log canonical and Q-factorial surfaces over excellent base schemes.2010 Mathematics Subject Classification. 14E30.

Shear and elongational flow measurements on polystyrene melts reinforced with small particles were carried out. The influences of loading level, particle size and surface treatment on shear viscosity, principal normal stress difference, and elongational viscosity were discussed. These systems exhibited yield values for both shear and elongational flow. Experimental values for the ratio of the tensile to the shear yield stress give satisfactory agreement with the predictions of the von Mises yield criterion. The yield value appears to increase with decreasing particle size and may be varied with surface treatment. The principal normal stress difference at fixed shear stress decreases with volume loading of particulates. The results are interpreted in terms of a system forming a gel due to interparticle forces, which is disrupted by a critical distortional strain energy.

Abstract. We discuss the birational geometry of singular surfaces in positive characteristic. More precisely, we establish the minimal model program and the abundance theorem for Q-factorial surfaces and for log canonical surfaces. Moreover, in the case where the base field is the algebraic closure of a finite field, we obtain the same results under much weaker assumptions.

We prove that if (X, A + B) is a pair defined over an algebraically closed field of positive characteristic such that (X, B) is strongly F -regular, A is ample and K X + A + B is strictly nef, then K X + A + B is ample. Similarly, we prove that for a log pair (X, A + B) with A being ample and B effective, K X + A + B is big if it is nef and of maximal nef dimension. As an application, we establish a rationality theorem for the nef threshold and various results towards the minimal model program in dimension three in positive characteristic.