On Kawamata-Viehweg Vanishing for Surfaces of del Pezzo type over imperfect fields

We prove the Kawamata-Viehweg vanishing theorem for surfaces of del Pezzo type over imperfect fields of characteristic $p>5$. As a consequence, we deduce the Grauert-Riemenschneider vanishing theorem for excellent divisorial log terminal threefolds whose closed points have (possibly imperfect) residue fields of positive characteristic $p>5$. Finally, under this setup, we show that three-dimensional klt singularities are rational.