Abstract
Let Y be a smooth complex projective variety of dimension \(n \ge 2\) endowed with a finite morphism \(\phi :Y \rightarrow {\mathbb {P}}^n\) of degree 3, and suppose that Y, polarized by some ample line bundle, is a scroll over a smooth variety X of dimension m. Then \(n \le 3\) and either \(m=1\) or 2. When \(m=1\), a complete description of the few varieties Y satisfying these conditions is provided. When \(m=2\), various restrictions are discussed showing that in several instances the possibilities for such a Y reduce to the single case of the Segre product \(\mathbb P^2 \times {\mathbb {P}}^1\). This happens, in particular, if Y is a Fano threefold as well as if the base surface X is \({\mathbb {P}}^2\).
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Acknowledgements
Both authors are members of INdAM (G.N.S.A.G.A.). The first author is grateful to Prof. Jack Huizenga for useful correspondence on the range of applicability of the ampleness criterion in [16, Theorem 5.1] in connection with a vector bundle corresponding to (6.1.1). Thanks are due to the referee for useful comments which helped to improve the presentation of the paper.
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Dedicated to Enrique Arrondo on the occasion of his 60th birthday.
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Lanteri, A., Novelli, C. Triple solids and scrolls. Rend. Circ. Mat. Palermo, II. Ser (2024). https://doi.org/10.1007/s12215-024-01020-8
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DOI: https://doi.org/10.1007/s12215-024-01020-8