Surfaces in \({\mathbb {P}^{4}}\) fibered in cubics. (English) Zbl 1181.14054
Let \(S\) be a smooth complex projective rational surface in \(\mathbb P^4\) ruled by cubics. Ph. Ellia [J. Pure Appl. Algebra 152, No. 1–3, 83–88 (2000; Zbl 0971.14033)] showed that the pair \((d,g)\) consisting of the degree and the sectional genus of \(S\) is confined to 5 possibilities. In the paper under review the authors obtain a complete classification: \((d,g)\) can only be \((5,2)\) or \((6,3)\), \(S\) is \(\mathbb F_1\) blown up at \(7\) or \(9\) points respectively in general position, and the precise description of the linear system providing the embedding is given in both cases. The authors associate to \(S\) a curve \(\Gamma\) in a suitable compactification, \(X\), of the space of smooth rational cubic curves in \(\mathbb P^4\), inspired by a construction of G. Ellingsrud and S. A. Strømme [Math. Scand. 76, No. 1, 5–34 (1995; Zbl 0863.14033)]. Then they find suitable generators for \(A^1(X) \otimes \mathbb Q\) the degrees of \(\Gamma\) with respect to which have relevant geometric meaning for the surface \(S\). Combining this with the double point formula for surfaces in \(\mathbb P^4\) provides remarkable constraints on the blow-ups a birational morphism \(S \to \mathbb F_e\) factors through. This leads to the result.
Reviewer: Antonio Lanteri (Milano)
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