Relative Riemann - Zariski

Let k be an algebraically closed field andK be a finitely generated k-field. In the first half of the 20-th century, Zariski defined a Riemann variety RZK(k) associated to K as the projective limit of all projective k-models of K. Zariski showed that this topological space, which is now called a Riemann-Zariski (or Zariski-Riemann) space, possesses the following set-theoretic description: to give a point x ∈ RZK is equivalent to give a valuation ring Ox with fraction field K and such that k ⊂ Ox. The Riemann-Zariski space possesses a sheaf of rings O whose stalks are valuation rings of K as above. Zariski made extensive use of these spaces in his desingularization works. Let S be a scheme and U be a subset closed under generalizations, for example U = Sreg is the regular locus of S, or U = η is a generic point of S. In many birational problems one wants to consider only U -modifications S → S, i.e. modifications which do not modify U . Then it is natural to consider the projective limit S = RZU (S) of all U -modifications of S. It was remarked in [Tem2, §3.3] that working with such relative Riemann-Zariski spaces one can extend the P -modification results of [Tem2] to the case of general U and S, and this plan is realized in §2. In §2.2 we give a preliminary description of the space S, which is used in §2.3 to prove the first main result of the paper, the stable modification theorem 2.3.2 generalizing its analog from [Tem2]. Our improvement to the stable modification theorem [Tem2, 1.1] is in the control on the base change one has to perform in order to construct a stable modification of a relative curve C → S. Namely, we prove that in order to find a stable modification of a relative curve with semi-stable U -fibers it suffices to replace the base S with a U -etale covering. Though a very rough study of relative RZ spaces suffices for the proof of theorem 2.3.2, it seems natural to investigate these spaces deeper. Furthermore, the definition of relative Riemann-Zariski spaces can be naturally generalized to the case of an arbitrary morphism f : Y → X , and the case when f is a dominant point was already applied in [Tem1], so it is natural to investigate the relative RZ spaces attached to a morphism f : Y → X . We will see that under a very mild assumption that f is a separated morphism between quasi-compact quasi-separated schemes, one obtains a very specific description of the space RZY (X) which is similar to the classical case of RZK(k). Let us say that f is decomposable if it factors into a composition of an affine morphism Y → Z and a proper morphism Z → X . Actually, in §2.2 we study RZY (X) in the case of a general decomposable morphism because