The specialization index of a variety over a discretely valued field.

In my thesis, I try to explain the non-existence of rational points on certain varieties. Let R be a complete discrete valuation ring with algebraically closed residue field k, and denote by K its fraction field. Let X be a proper scheme over K. An important obstruction to the existence of a rational point on X is the index, the minimal positive degree of a zero-cycle on X. It is known that the index can be computed explicitly, only using data coming from the special fiber of a proper regular model of X. Moreover, the index of a smooth, proper, geometrically connected variety over C((t)) (the formal Laurent series over the complex numbers) with trivial coherent cohomology is equal to one.

Nicaise introduced a new invariant, the specialization index, which is a closer approximation to the existence of a rational point. Assuming a suitable form of resolution of singularities, we provide an explicit formula for the specialization index in terms of the special fiber of an snc-model. There are examples of curves where the index equals one but the specialization index is different from one, and thus explains the absence of a rational point. The main result that the specialization index of a smooth, proper, geometrically connected C((t))-variety with trivial coherent cohomology is equal to one. In the proof we need equivariant semi-stable reduction over a discrete valuation ring, of which there exists no detailed proof in literature. I give a detailed proof.

Furthermore, I use logarithmic geometry and give a formula for the specialization index of a log smooth model in terms of its log structure. In characteristic zero, this formula specializes to the formula already given before. I use this to give a relation between the specialization index of the fibered product of two varieties over K and the specialization indices of the individual varieties. Finally, I construct an example, which shows that this inequality can also be strict.