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Weak toroidalization over non-closed fields. (English) Zbl 1279.14020

The authors prove that a dominant morphism of algebraic varieties over a field \(k\) of characteristic zero can be “improved” by suitable projective modifications of the domain and codomain. More precisely, if \(f:X \to B\) is a dominant morphism, there are projective, birational morphisms \(m_X:X' \to X\) and \(m_B:B' \to B\), with \(W'\) and \(B'\) smooth, quasiprojective varieties, open toroidal embeddings \(U_{X'} \subset X'\) and \(U_{B'} \subset B'\), so that the induced birational map \(f':X' \to B'\) is a toroidal morphism. Moreover, if \(Z\) is a proper closed subset of \(X'\) then \(Z' = m_X ^{-1}(Z)\) is a strict normal crossings divisor of \(X'\), disjoint with \(U_{X'}\), and \(U_{X'}\) is isomorphic to \(m_{X'}(U_X)\). A similar result, but also assuming \(k\) algebraically closed, was proved in [D. Abramovich and K. Karu, Invent. Math. 139, No. 2, 241–273 (2000; Zbl 0958.14006)], although for some applications the algebraically closed condition is too restrictive. The present demonstration is based on that of the mentioned paper, but sometimes special care is necessary to show that certain objects are defined over \(k\) itself and not some algebraic extension of it.
The proof begins with a series of reductions that allow us to assume, among other things, that \(X\) and \(B\) are projective and normal. Then one uses an inductive argument (on the relative dimension of \(X\) over \(B\)). Namely, the morphism \(f\) is factored as \(X \to P \to B\), where \(\dim P= \dim X -1\). After further work one may assume that the morphism \(X \to P\) defines a semistable family of curves (the geometric fibers are connected nodal curves). This fact and the inductive hypothesis of \(P \to B\) allow the authors, after considerable work, to finish the proof. The base of the induction (i.e., the case where the relative dimension of \(X\) over \(B\) is equal to zero) is not trivial. In this case, after several reductions, the authors are able to use a variant of Abhyankar’s Lemma. Although eventually the morphisms \(m_X\) and \(m_B\) are birational, in the course of the proof they use alterations (proper, generally finite morphisms). They go back to birational morphisms by taking quotients under the action of suitable finite groups.
The article is fairly self-contained. The authors briefly review a number a number of necessary notions and results, which facilitates the reading.

MSC:

14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14B25 Local structure of morphisms in algebraic geometry: étale, flat, etc.
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14H10 Families, moduli of curves (algebraic)

Citations:

Zbl 0958.14006
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References:

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