Speiser, Robert Transversality theorems for families of maps. (English) Zbl 0683.14003 Algebraic geometry, Proc. Conf., Sundance/Utah 1986, Lect. Notes Math. 1311, 235-252 (1988). [For the entire collection see Zbl 0635.00006.] Let f: \(X\to Z\) and g: \(Y\to Z\) be morphisms of schemes. Let \(W=X\times_ ZY\). We say that f and g meet properly (resp. are transverse) if for each (x,y) of Z with \(z=f(x)=g(y)\), we have \(\dim_{(x,y)}W=\dim_ xX+\dim_ yY-\dim_ zZ\) (resp. the tangent spaces \(T_ xX\) and \(T_ yY\) span \(T_ zZ)\). In this article, the deformation problem of transversality is considered in the following setup: Let X, Y, Z and S be smooth varieties defined over an algebraically closed field, and we are given morphisms \(f:\quad X\to Z,\) \(g:\quad Y\to Z\) and a smooth morphism \(\pi:\quad X\to S.\) Let \(W=X\times_ ZY\). Then, for which \(s\in S\), \(\pi^{-1}(s)=X_ s\) together with the morphism \(X_ s\hookrightarrow X\to Z\) is transverse to g? The present work is closely connected to the works of S. Kleiman [Compos. Math. 28, 287-297 (1974; Zbl 0288.14014)] and D. Laksov [ibid. 30, 273-292 (1975; Zbl 0306.14022)]. Reviewer: M.Miyanishi Cited in 3 ReviewsCited in 6 Documents MSC: 14D15 Formal methods and deformations in algebraic geometry 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 14E05 Rational and birational maps Keywords:proper intersection; transverse morphisms of schemes; deformation problem of transversality Citations:Zbl 0635.00006; Zbl 0288.14014; Zbl 0306.14022 PDFBibTeX XML