Weak del Pezzo surfaces with global vector fields

We classify smooth weak del Pezzo surfaces with global vector fields over an arbitrary algebraically closed field k of arbitrary characteristic p 0. We give a complete description of the configuration of .1/and .2/-curves on these surfaces and calculate the identity component of their automorphism schemes.It turns out that there are 53 distinct families of such surfaces if p 2; 3, while there are 61 such families if p D 3 and 75 such families if p D 2. Each of these families has at most one moduli.As a byproduct of our classification, it follows that weak del Pezzo surfaces with nonreduced automorphism schemes exist over k if and only if p 2 f2; 3g.