Families of affine ruled surfaces: existence of cylinders (Q2828018)

Let \(X\) and \(Y\) be normal complex algebraic varieties. A dominant morphism \(p : X \to Y\) is called an \(\mathbb A^1\)-fibration if the general closed fibers of \(p\) are isomorphic to \(\mathbb A^1\). If \(X\) is a smooth affine algebraic surface, it is well-known that \(X\) has negative logarithmic Kodaira dimension if and only if \(X\) is \(\mathbb A^1\)-ruled, i.e., \(X\) contains a Zariski open dense subset \(U\) of the form \(U\cong B\times \mathbb A^1\) for a smooth curve \(B\) ([\textit{M. Miyanishi} and \textit{T. Sugie}, J. Math. Kyoto Univ. 20, 11--42 (1980; Zbl 0445.14017)], [\textit{T. Fujita}, Proc. Japan Acad., Ser. A 55, 106--110 (1979; Zbl 0444.14026)]). The projection \(U\cong B\times \mathbb A^1\to B\) extends to an \(\mathbb A^1\)-fibration \(\rho : X \to C\) where \(C\) is an open subset of a smooth projective model of \(B\). The \(\mathbb A^1\)-fibration \(\rho : X \to C\) is called of affine type (resp. complete type) if the base curve \(C\) is affine (resp. complete).NEWLINENEWLINEAn algebraic variety \(X\) is called \(\mathbb A^1\)-uniruled if there exists a dense subset \(U\) such that each point \(x\) of \(U\) belongs to the image of a non-constant morphism \(\mathbb A^1 \to X\). By the authors [Bull. Soc. Math. Fr. 143, No. 2, 383--401 (2015; Zbl 1327.14196)], it is known that there exist smooth affine varieties of dimension \(\geq 3\), which are \(\mathbb A^1\)-uniruled but not \(\mathbb A^1\)-ruled. One of such examples is a smooth affine threefold \(X\) with a flat morphism \(f : X \to \mathbb A^1=\mathrm{Spec} \mathbb C[t]\) such that general fibers have negative log Kodaira dimension and admit \(\mathbb A^1\)-fibrations of complete type only, whereas the generic fiber does not admit any \(\mathbb A^1\)-fibration over \(\mathbb C(t)\). Hence the threefold \(X\) has no \(\mathbb A^1\)-fibration. However, \(X\) has an \(\mathbb A^1\)-fibration after taking a finite covering of \(\mathbb A^1\), namely, there is a finite covering \(T\) of \(\mathbb A^1\) such that the normalization of \(X\times_{\mathbb A^1}T\) has an \(\mathbb A^1\)-fibraion [\textit{R. V. Gurjar} et al., in: Automorphisms in birational and affine geometry. Papers based on the presentations at the conference, Levico Terme, Italy, October 29 -- November 3, 2012. Cham: Springer. 327--361 (2014; Zbl 1326.14146)]. The main result (Theorem 7) of this paper shows that this holds in general: Let \(X\) and \(S\) be normal algebraic varieties and let \(f: X \to S\) a dominant morphism such that general fibers are smooth \(\mathbb A^1\)-ruled affine surfaces. Then there exist a dense open subset \(S_*\) of \(S\), an étale finite morphism \(T \to S_*\), and a normal \(T\)-scheme \(h : Y \to T\) such that \(f_T: X\times_{S_*}T \to T\) factors as \(f_T=h \circ \rho\) where \(\rho : X\times_{S_*}T \to Y\) is an \(\mathbb A^1\)-fibration. In the last two sections, the authors study the case where the general fibers are irrational \(\mathbb A^1\)-ruled surfaces. In that case, there is an open subset \(S_*\) of \(S\) such that the restriction of \(f\) to \(X_*=X\times_SS_*\) factors as a composite \(h \circ \rho\) for an \(\mathbb A^1\)-fibration \(\rho : X_* \to Y\) and a normal \(S_*\)-scheme \(h : Y \to S_*\) (Theorem 10). The \(\mathbb A^1\)-fibration \(\rho: X_* \to Y\) can be viewed as the MRC-fibration (maximally rationally connected fibration) of a relative smooth projective model \(\overline X\) of \(X\) over \(S\).