Deformations of $\mathbb{A}^1$-cylindrical varieties

Adrien Dubouloz (1) , Takashi Kishimoto (2)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]
2 Department of Mathematics, Faculty of Science, Saitama University

An algebraic variety is called $\mathbb{A}^{1}$-cylindrical if it contains an $ \mathbb{A}^{1}$-cylinder, i.e. a Zariski open subset of the form $Z\times\mathbb{A}^{1}$ for some algebraic variety $Z$. We show that the generic fiber of a family $f:X\rightarrow S$ of normal $\mathbb{A}^{1}$-cylindrical varieties becomes $\mathbb{A}^{1}$-cylindrical after a finite extension of the base. Our second result is a criterion for existence of an $\mathbb{A}^{1}$-cylinder in $X$ which we derive from a careful inspection of a relative Minimal Model Program ran from a suitable smooth relative projective model of $X$ over $S$.

Mots clés: Minimal Model Program; $\mathbb{A}^{1}$-cylinder; Deformation; Uniruled varieties

Domaines: Géométrie algébrique [math.AG]

Adrien Dubouloz, Takashi Kishimoto. Deformations of $\mathbb{A}^1$-cylindrical varieties. Mathematische Annalen, 2019, 373 (3-4), pp.1135-1149. ⟨10.1007/s00208-018-1774-9⟩. ⟨hal-01622447⟩