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Nonrational hypersurfaces. (English) Zbl 0839.14031

In this very interesting paper there are developed some general ideas to obtain examples of Fano varieties, non-rational and not even ruled. – Complete proofs, here only sketched, will appear elsewhere. The results are valid for “very general” hypersurfaces, that is they hold for hypersurfaces corresponding to a point in the complement of countably many closed subvarieties in the space of all hypersurfaces. The method is by degeneration and it works thanks to a previous result of Matsusaka, which compels to pass to positive characteristic. (So that concrete examples are over \(\mathbb{Q})\).
One of the main results is the following \((\lceil x\rceil\) denotes the smallest integer \(\geq x)\).
Let \(X_d \subset \mathbb{P}^{n + 1}\) be a very general hypersurface over \(\mathbb{C}\) of degree \(d\); if \(d \geq 2^{\lceil (n + 3)/3 \rceil}\) then \(X_d\) is not ruled; if \(d \geq 3^{\lceil (n + 3)/4 \rceil}\) then \(X\) is not birational to a conic bundle. Assume \(d = n + 1\). Let \(Y\) be a variety of dimension \(n - 1\) and \(\varphi : Y \times \mathbb{P}^1 \to X\) a dominant map. Then \(\deg \varphi\) is divisible by every prime less than \(\sqrt n\).

MSC:

14J45 Fano varieties
14J70 Hypersurfaces and algebraic geometry
14M20 Rational and unirational varieties
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