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Birational self-maps of threefolds of (un)-bounded genus or gonality. (English) Zbl 1489.14025

Let \(X\) be a smooth projective complex algebraic variety. Denote by \(\text{Bir}(X)\) the group of birational self-maps of \(X\). For any \(\varphi\in \text{Bir}(X)\), define the genus \(g(\varphi)\) (resp. the gonality \(\text{gon}(\varphi)\)) by the maximnum of \(g(C)\) (resp. \(\text{gon}(C)\)) such that \(\varphi\) contracts a divisor of \(X\) birational to \(\mathbb{P}^1\times C\).
This paper characterizes whether all elements in \(\text{Bir}(X)\) have bounded genus and gonality by the birational structure of \(X\) when \(\dim X=3\). It shows that if \(\dim X=3\), all elements in \(\text{Bir}(X)\) have bounded genus if and only if \(X\) is not birational to a conic bundle nor a del Pezzo fibration of degree \(3\); all elements in \(\text{Bir}(X)\) have bounded gonality if and only if \(X\) is not birational to a conic bundle.

MSC:

14E30 Minimal model program (Mori theory, extremal rays)
14J45 Fano varieties
14E05 Rational and birational maps
14E07 Birational automorphisms, Cremona group and generalizations
14J30 \(3\)-folds

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