Zero cycles on del Pezzo surfaces over local fields (Q1075400)

Let \(A_ 0(X)\) be the degree zero zero-cycles in the Chow ring of X. By studying \(H^ 1(Pic\)(X\({}_ E))\) the author proves: Theorem 1. Let X be a del Pezzo surface of degree \(d\geq 5\), defined over a local field k. Then \(A_ 0(X)=0.\) Theorem 2. A fine moduli space for the set of marked split nonsingular del Pezzo surfaces is \(M={\mathbb{P}}^ 2-C\), where C is the union of lines through four fixed points. To a point (a;b;c)\(\in M\) we associate the surface \(bsv=ar(u-t)+(e-b)su\), \(btv=cu(r-s)+(a-b)rt.\) Theorem 3. Let E/k be an unramified quadratic extension of local fields. Let X be a degree 4 del Pezzo k-surface split by E. Then either \(A_ 0(X)=0\) or X is a type IV surface with bad reduction.