A computation of invariants of a rational self-map

Amerik, Ekaterina

doi:10.5802/afst.1211

Cited by 21 publications

(150 citation statements)

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“…Y ). As X is general, lines which are special in X are parameterized by a smooth surface Σ sp ⊂F (X) (see [6]). …”

Section: Proof Of Proposition 23 Let Us Say That a Linementioning

confidence: 99%

“…When X contains no plane, the indeterminacy locus of φ is exactly the surface Σ sp , along which the plane P l above is not unique. Furthermore, the indeterminacies of the map φ are solved after blowing-up the surface Σ sp , and the induced morphismφ: F (X)!F (X) is finite if X is general (see [6]). Note that the condition on a line l ⊂Y to being good will be implied by the slightly stronger fact that l is non-special in X (so φ is well defined at [l ]) and for no point…”

Section: Proof Of Proposition 23 Let Us Say That a Linementioning

confidence: 99%

“…Let us say that a class γ ∈Hdg Coming back to the proof of Proposition 2.3, it is clear that for any hyperplane section Y of X, there is a line contained in Y which is non-special in X. Indeed, the surface Σ sp of lines which are special in X is irreducible and not contained in the surface of lines in Y because it is smooth connected and not Lagrangian, see [6]; thus it can intersect F (Y ) only along a proper subset. Next, assume to the contrary that there is no good line in Y.…”

Section: Proof Of Proposition 23 Let Us Say That a Linementioning

confidence: 99%

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A hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-fold

2017

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“…Y ). As X is general, lines which are special in X are parameterized by a smooth surface Σ sp ⊂F (X) (see [6]). …”

Section: Proof Of Proposition 23 Let Us Say That a Linementioning

confidence: 99%

Section: Proof Of Proposition 23 Let Us Say That a Linementioning

confidence: 99%

Section: Proof Of Proposition 23 Let Us Say That a Linementioning

confidence: 99%

See 1 more Smart Citation

A hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-fold

2017

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“…Indeed, we may assume by induction on l that φ l−1 is generically defined along Σ b . Then φ l−1 (Σ b ) must be a surface, as proved in [1]. If φ l were not generically defined along Σ b , then φ l−1 (Σ b ) would be contained in the indeterminacy locus of φ.…”

Section: Proposition 21 Under This Assumption the Zariski Closure Omentioning

confidence: 81%

“…If φ l were not generically defined along Σ b , then φ l−1 (Σ b ) would be contained in the indeterminacy locus of φ. But this indeterminacy locus is a surface of general type ( [1]), hence cannot be dominated by a surface which is birationally equivalent to an abelian surface.…”

Section: Proposition 21 Under This Assumption the Zariski Closure Omentioning

confidence: 99%

Potential density of rational points on the variety of lines of a cubic fourfold

Amerik

Voisin

2008

Duke Math. J.

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Recall that for a variety X defined over a field k, rational points are said to be potentially dense in X if for some finite extension k ′ of k, X(k ′ ) is Zariski dense in X. For example, this is the case if X is a rational or a unirational variety.If k is a number field, a conjecture of Lang and Vojta, proved in dimension 1 by Faltings, but still open even in dimension two, predicts that varieties of general type never satisfy potential density over k. On the contrary, it is generally expected that when the canonical bundle K X is negative (i.e. X is a Fano variety) or trivial, rational points are always potentially dense in X (see for example [9] for a recent and precise conjecture).In the Fano case, there are many examples confirming this. First of all, many Fano varieties are unirational (in dimension ≤ 2, all of them are even rational over a finite extension of the definition field). Furthermore, Fano threefolds such as a three-dimensional quartic or a "double Veronese cone" (a hypersurface of degree 6 in P(1, 1, 1, 2, 3)) are birational to elliptic fibrations over P 2 , which makes it possible to prove potential density (see [13]).If the canonical bundle is trivial, the known examples are much less convincing. It is well-known that rational points on abelian varieties are potentially dense. But the simply-connected case remains largely unsolved.A number of results concerning potential density of rational points on K3 surfaces defined either over a number field or a function field appeared in the last years. Bogomolov and Tschinkel [6] proved potential density of rational points for K3 surfaces admitting an elliptic pencil, or an infinite automorphisms group. Remark that these K3 surfaces are rather special; in particular, their geometric Picard number is never equal to 1, whereas it is equal to 1 for a general projective K3.The case of K3-surfaces defined over a (complex) function field has been solved for certain types of families by Hassett and Tschinkel [14]. They prove in particular the existence of K3-surfaces defined over C(t), whose geometric Picard group is equal to Z, and which satisfy potential density. Their method can likely be adapted to produce examples over Q(t).However, no example of K3 surface defined over a number field, and with geometric Picard group equal to Z is known to satisfy potential density.In this paper, we consider the case of higher dimensional varieties which are as close as possible to K3 surfaces, namely Fano varieties of lines F of cubic 4-folds X. These are 4-dimensional varieties with trivial canonical bundles, and which even possess a non degenerate holomorphic 2-form which will play an important role in our study. Their similarity with K3 surfaces is shown by the work of Beauville and 1

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Invertible Dynamics on Blow-ups of $${\mathbb {P}}^{k}$$

Bedford

2015

Springer Proceedings in Mathematics &Amp; Statistics

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A computation of invariants of a rational self-map

Cited by 21 publications

References 7 publications

A hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-fold

A hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-fold

Potential density of rational points on the variety of lines of a cubic fourfold

Invertible Dynamics on Blow-ups of $${\mathbb {P}}^{k}$$

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