- Equivariant extensions of $\mathbb{G}_a$-torsors over punctured surfaces, with Isac Hedén and Takashi Kishimoto,
Ann. Sc. Norm. Super. Pisa Cl. Sci. (to appear)) arXiv:1707.08768.
Abstract
Motivated by the study of the structure of algebraic actions the additive group on affine threefolds $X$,
we consider a special class of such varieties whose algebraic quotient morphisms $X \rightarrow X/\!/\mathbb{G}_a$
restrict to principal homogeneous bundles over the complement of a smooth point of the quotient.
We establish basic general properties of these varieties and construct families of examples illustrating their rich geometry.
In particular, we give a complete classification of a natural subclass consisting of threefolds $X$ endowed with proper
$\mathbb{G}_a$-actions, whose algebraic quotient morphisms $\pi : X \rightarrow X/\!/\mathbb{G}_a$ are surjective
with only isolated degenerate fibers, all isomorphic to the affine plane $\mathbb{A}^2$ when equipped with their reduced structures.
- Cylinders in Mori Fiber Spaces: forms of the quintic del Pezzo threefold, with Takashi Kishimoto,
Ann. Inst. Fourier (to appear), arXiv:1805.05679.
Abstract
Motivated by the general question of existence of open $\mathbb{A}^1$-cylinders in higher dimensional projective varieties,
we consider the case of Mori Fiber Spaces of relative dimension three, whose general closed fibers are isomorphic to the
quintic del Pezzo threefold $V_5$, the smooth Fano threefold of index two and degree five.
We show that the total spaces of these Mori Fiber Spaces always contain relative $\mathbb{A}^2$-cylinders, and we characterize
those admitting relative $\mathbb{A}^3$-cylinders in terms of the existence of certain special lines in their generic fibers.
- Algebraic models of the real affine plane, with Jérémy Blanc,
EPIGA Volume 2 (2018), Article Nr. 14, arXiv:1708.08058v2.
Abstract
We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that allows to distinguish smooth real algebraic
surfaces up to birational diffeomorphism. As an application, we construct infinite families of smooth rational real
algebraic surfaces with trivial homology groups, whose real loci are diffeomorphic to $\mathbb{R}^2$, but which are
pairwise not birationally diffeomorphic. There are thus infinitely many non-trivial models of the real affine plane,
contrary to the compact case.
- Deformations of $\mathbb{A}^1$-cylindrical varieties, with Takashi Kishimoto,
Math. Ann. (to appear), arXiv:1710.09108.
Abstract
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.
- Super-rigid affine Fano varieties, with Ivan Cheltsov and Jihun Park,
Compositio Math. vol 154, Issue 11 (2018), 2463-2484, arxiv:1712.09148.
Abstract
We study a wide class of affine varieties, which we call affine Fano varieties.
By analogy with birationally super-rigid Fano varieties, we define super-rigidity for affine Fano varieties,
and provide many examples and non-examples of super-rigid affine Fano varieties.
- The Jacobian Conjecture fails for pseudo-planes, with Karol Palka,
Advances in Math. Volume 339 (2018), 248-284, arXiv:1701.01425.
Abstract
A smooth complex variety satisfies the Generalized Jacobian Conjecture if all its etale endomorphisms are proper.
We study the equivariant version of the conjecture for $\mathbb{Q}$-acyclic surfaces of negative Kodaira dimension and
infinite algebraic groups. We show that it holds for groups other than $\mathbb{C}^*$, and for $\mathbb{C}^*$ we classify
counterexamples relating them to Belyi-Shabat polynomials. Taking universal covers we get rational simply connected $\mathbb{C}^*$-surfaces
of negative Kodaira dimension which admit non-proper $\mathbb{C}^*$-equivariant etale endomorphisms.
We prove that for every integers $r\geq 1$, $k\geq2$, the $\mathbb{Q}$-acyclic rational hyperplane $u(1+u^rv)=w^k$, which has fundamental
group $\mathbb{Z}/k\mathbb{Z}$ and negative Kodaira dimension, admits families of non-proper etale endomorphisms of arbitrarily
high dimension and degree, whose members remain different after dividing by the action of the automorphism group by left and right composition.
- Cylinders in del Pezzo fibrations, with Takashi Kishimoto,
Israël J. Math. Volume 225, Issue 2 (2018), 797-815, arXiv:1607.00840.
Abstract
We show that a del Pezzo fibration $\pi:V\rightarrow W$ of degre
$d$ contains a vertical open cylinder, that is, an open subset whose
intersection with the generic fiber of $\pi$ is isomorphic to $Z\times\mathbb{A}_{K}^{1}$
for some quasi-projective variety $Z$ defined over the function field
$K$ of $W$, if and only if $d\geq5$ and $\pi:V\rightarrow W$ admits
a rational section. We also construct twisted cylinders in total spaces
of threefold del Pezzo fibrations $\pi:V\rightarrow\mathbb{P}^{1}$
of degree $d\leq4$.
- Fake Real Planes: exotic affine algebraic models of $\mathbb{R}^{2}$ , with Frédéric Mangolte,
Selecta Math., 23(3), 2017, 1619-1668. arXiv:1507.01574.
Abstract
We study real rational models of the euclidean affine plane $\mathbb{R}^{2}$ up to isomorphisms and up to birational diffeomorphisms. The analogous study in
the compact case, that is the classification of real rational models of the real projective plane $\mathbb{R}\mathbb{P}^{2}$ is well known: up to
birational diffeomorphisms, $\mathbb{P}^{2}(\mathbb{R})$ is the only model. A fake real plane is a smooth geometrically integral surface $S$ defined over
$\mathbb{R}$ not isomorphic to $\mathbb{A}^2_\mathbb{R}$, whose real locus $S(\mathbb{R})$ is diffeomorphic to $\mathbb{R}^2$ and such that the complex
surface $S_\mathbb{C}(\mathbb{C})$ has the rational homology type of $\mathbb{A}^2_\mathbb{C}$. We prove that fake planes exist by giving many
examples and we tackle the question: does there exist fake planes $S$ such that $S(\mathbb{R})$ is not birationally diffeomorphic to $\mathbb{A}^2_\mathbb{R}(\mathbb{R})$ ?
- Affine lines in the complement of a smooth plane conic, with Julie Decaup
Boll Unione Mat Ital, Volume 11, Issue 1 (2018), 39–54. arXiv:1611.03248.
Abstract
We classify closed curves isomorphic to the affine line in the complement of a smooth rational projective plane conic $Q$.
Over a field of characteristic zero, we show that up to the action of the subgroup of the Cremona group of
the plane consisting of birational endomorphisms restricting to biregular auto-morphisms outside $Q$,
there are exactly two such lines: the restriction of a smooth conic osculating $Q$ at a rational point and the restriction
of the tangent line to $Q$ at a rational point.
In contrast, we give examples illustrating the fact that over fields of positive characteristic, there exist exotic closed
embeddings of the affine line in the complement of $Q$.
We also determine an explicit set of birational endomorphisms of the plane whose restrictions generates the
automorphism group of the complement of $Q$ over a field of arbitrary characteristic.
- Families of exotic affine 3-spheres,
European Journal of Mathematics, 4(1) (2018), 212-222. arXiv:1610.01409.
Abstract
We construct algebraic families of exotic affine 3-spheres, that is, smooth affine threefolds diffeomorphic to a non-degenerate smooth
complex affine quadric of dimension 3 but non algebraically isomorphic to it.
We show in particular that for every smooth topologically contractible affine surface S with trivial automorphism group,
there exists a canonical smooth family of pairwise non isomorphic exotic affine 3-spheres parametrized by the closed points of S.
- Affine surfaces with isomorphic $\mathbb{A}^2$-cylinders,
Kyoto J. Maths (to appear), arXiv:1507.05802.
Abstract
We construct families of smooth affine surfaces with pairwise non isomorphic $\mathbb{A}^1$-cylinders but whose $\mathbb{A}^2$-cylinders are all isomorphic.
These arise as complements of cuspidal hyperplane sections of smooth projective cubic surfaces.
- Families of $\mathbb{A}^1$-contractible affine threefolds, with Jean Fasel,
Algebraic Geometry 5 (1) (2018) 1-14. arXiv:1512.01933.
Abstract
We provide families of affine threeAlgebraic Geometry 5 (1) (2018) 1-14.
- Explicit biregular/birational geometry of affine threefolds: completions of $\mathbb{A}^3$ into del pezzo fibrations and Mori conic bundles, with Takashi Kishimoto,
Adv. Stud. Pure Math. 75, Mathematical Society of Japan, Tokyo, 2017, 49-71. arXiv:1508.01792.
Abstract
We study certain pencils $\overline{f}:\mathbb{P}\dashrightarrow\mathbb{P}^{1}$
of del Pezzo surfaces generated by a smooth del Pezzo surface $S$ of degree less or equal to $3$ anti-canonically embedded into a weighted projective space $\mathbb{P}$
and an appropriate multiple of a hyperplane $H$. Our main observation is that every minimal model program relative to the morphism
$\tilde{f}:\tilde{\mathbb{P}}\rightarrow\mathbb{P}^{1}$ lifting $\overline{f}$ on a suitable resolution $\sigma:\tilde{\mathbb{P}}\rightarrow\mathbb{P}$
of its indeterminacies preserves the open subset $\sigma^{-1}(\mathbb{P}\setminus H)\simeq\mathbb{A}^{3}$.
As an application, we obtain projective completions of $\mathbb{A}^{3}$ into del Pezzo fibrations over $\mathbb{P}^{1}$
of every degree less or equal to $4$. We also obtain completions of $\mathbb{A}^{3}$ into Mori conic bundles, whose restrictions to $\mathbb{A}^{3}$
are twisted $\mathbb{A}_{*}^{1}$-fibrations over $\mathbb{A}^{2}$.
- Affine-ruled varieties without the Laurent cancellation property, with Pierre-Marie Poloni,
Bull. London Math. Soc. (2016) 48 (5): 822-834. arXiv:1509.07803.
Abstract
We describe a method to construct hypersurfaces of the complex affine $n$-space with isomorphic $\mathbb{ℂ}^∗$-cylinders. Among these hypersurfaces,
we find new explicit counterexamples to the Laurent Cancellation Problem, i.e. hypersurfaces that are non isomorphic, although their $\mathbb{ℂ}^∗$-cylinders
are isomorphic as abstract algebraic varieties. We also provide examples of non isomorphic varieties $X$ and $Y$ with isomorphic cartesian squares $X×X$ and $Y×Y$.
- Rationally integrable vector fields and rational additive group actions, with Alvaro Liendo,
International J. of Math. Vol. 27, No. 8 (2016) 1650060. arXiv:1409.5878.
Abstract
We characterize rational actions of the additive group on algebraic varieties defined over a
field of characteristic zero in terms of a suitable integrability property of their associated velocity vector fields. This extends the classical
correspondence between regular actions of the additive group on affine algebraic varieties and the so-called locally nilpotent derivations of their
coordinate rings. This leads in particular to a complete characterization of regular additive group actions on semi-affine varieties in terms of their
associated vector fields. Among other applications, we review properties of the rational counter-part of the Makar-Limanov invariant for affine
varieties and describe the structure of rational homogeneous additive group actions on toric varieties.
- On the cancellation problem for algebraic tori,
Ann. Inst. Fourier Vol. 66 (6) (2016), 2621-2640. arXiv:1412.2213.
Abstract
We address a variant of Zariski Cancellation Problem, asking whether two varieties which become
isomorphic after taking their product with an algebraic torus are isomorphic themselves. Such cancellation property
is easily checked for curves, is known to hold for smooth varieties of log-general type by virtue of a result of Iitaka-
Fujita and more generally for non $\mathbb{A}^1_*$-uniruled varieties. We show in contrast that for smooth affin
e factorial $\mathbb{A}^1_*$-ruled varieties, cancellation fails in any dimension bigger or equal to two.
- Real Frontiers of Fake Planes, with Frédéric Mangolte,
European Journal of Mathematics, 2(1) (2016), 140-168. arXiv:1508.07695.
Abstract
In arXiv:1507.01574, we define and partially classify fake real planes, that is, minimal complex surfaces
with conjugation whose real locus is diffeomorphic to the euclidean real plane $\mathbb{R}^2$. Classification results are given up to biregular
isomorphisms and up to birational diffeomorphisms. In this note, we describe in an elementary way numerous examples of fake real planes
and we exhibit examples of such planes of every Kodaira dimension $\kappa\in \{-\infty,0,1,2 \}$ which are birationally diffeomorphic to $\mathbb{R}^2$.
- Families of affine ruled surfaces: existence of cylinders, with Takashi Kishimoto,
Nagoya Mathematical Journal, Vol 223 (1) (2016), 1-20. arXiv:1408.1328.
Abstract
We show that the generic fiber of a family of smooth $\mathbb{A}^1$-ruled affine surfaces
always carries an $\mathbb{A}^1$-fibration, possibly after a finite extension of the base. In the particular case where the general fibers of the
family are irrational surfaces, we establish that up to shrinking the base, such a family actually factors through an $\mathbb{A}^1$-fibration over
a certain scheme, induced by the MRC-fibration of a relative smooth projective model of the family. For affine threefolds fibered by irrational
$\mathbb{A}^1$-ruled surfaces, this induced $\mathbb{A}^1$-fibration can also be obtained from a relative Minimal Model Program applied to a relative
smooth projective model of the family.
- Flexible bundles over rigid affine surfaces,
Comment. Math. Helv. 90 (2015), 121-137. arXiv:1304.4189.
- Log-uniruled affine varieties without cylinder-like open subsets, with Takashi Kishimoto,
Bull. Soc. math. France, 143 (1), 2015, 383-401. arXiv:1212.0521.
- Complements of hyperplane sub-bundles in projective spaces bundles over $ \mathbb{P}^1 $,
Math. Ann. 361 (2015), no 1-2, 259–273. arXiv:1108.6209.
- Affine Surfaces With a Huge Group of Automorphisms, with Jérémy Blanc,
Int. Math. Res. Notices, 2015 (2), 422-459. arXiv:1302.3813.
- Automorphisms of open surfaces with irreducible boundary, with Stéphane Lamy,
Osaka J. Math. Volume 52, Number 3 (2015), 747-793. arXiv:1404.4838.
- Proper triangular $\mathbb{G}_a$-actions on $\mathbb{A}^4$ are translations, with David R. Finston and Imad Jaradat,
Algebra Number Theory 8 (2014), no. 8, 1959–1984. arXiv:1303.1032.
- On exotic affine 3-spheres, with David R. Finston,
J. Algebraic Geom. 23 (2014), no. 3, 445–469. arXiv:1106.2900.
- Proper twin-triangular $\mathbb{G}_a$-actions on $\mathbb{A}^4$ are translations, with David R. Finston,
Proc. Amer. Math. Soc. 142 (2014), no. 5, 1513–1526. arXiv:1109.6302.
- Equivariant triviality of quasi-monomial triangular $\mathbb{G}_a$-actions on $\mathbb{A}^4$, with David R. Finston and Imad Jaradat,
in "Automorphisms in Birational and Affine Geometry" Proceedings of conference in Levico Terme, Italy, October, 2012,
Springer Proceedings in Mathematics and Statistics, (2014).
- Automorphism groups of certain rational hypersurfaces in complex four-space, with Lucy Moser-Jauslin and Pierre-Marie Poloni,
in "Automorphisms in Birational and Affine Geometry" Proceedings of conference in Levico Terme, Italy, October, 2012,
Springer Proceedings in Mathematics and Statistics, (2014). arXiv:1309.7363.
- Affine open subsets in $\mathbb{A}^3$ without the cancellation property,
in "Commutative algebra and algebraic geometry" (CAAG-2010), 63–67, Ramanujan Math. Soc. Lect. Notes Ser., 17, Ramanujan Math. Soc., Mysore, 2013.
arXiv:1107.1968.
- Locally tame plane polynomial automorphisms , with Joost Berson, Jean-Philippe Furter and Stefan Maubach,
Journal of Pure and Applied Algebra 216 (2012) 149-153. arXiv:1011.0976.
- Non cancellation for smooth contractible threefolds, with Lucy Moser-Jauslin and Pierre-Marie Poloni,
Proc. Amer. Math. Soc. 139 (2011) 4273-4284. arXiv:1004.4723.
- Automorphisms of $\mathbb{A}^1$-fibered surfaces, with Jérémy Blanc,
Trans. Amer. Math. Soc. 363 (2011), 5887-5924. arXiv:0906.3623v1.
- Inequivalent embeddings of the Koras-Russell cubic threefold, with Lucy Moser-Jauslin and Pierre-Marie Poloni,
Michigan Mathematical Journal, Volume 59, no. 3 (2010), p. 679-694. arXiv:0903.4278v1.
- The cylinder over the Koras-Russell cubic threefold has a trivial Makar-Limanov invariant,
Transformation Groups vol 14, no 3 (2009), p. 531-539. arXiv:0807.4085.
- On a class of Danielewski surfaces in affine 3-space , with Pierre-Marie Poloni,
Journal of Algebra 321 (2009), p.1797-1812. arXiv:math/0602549.
- Additive group actions on Danielewski varieties and the Cancellation Problem,
Math. Z. 255 (2007), no. 1, p. 77-93. arXiv:math/0507505v1.
- Embeddings of Danielewski surfaces in affine spaces,
Commentarii Mathematici Helvetici vol 81, no. 1, (2006), p. 49-73. arXiv:math/0403208.
- Danielewski-Fieseler surfaces,
Transformation Groups vol. 10, no. 2, (2005), p. 139-162. arXiv:math/0401225.
- Completions of normal affine surfaces with a trivial Makar-Limanov invariant,
Michigan Mathematical Journal vol. 52, no. 2, (2004), p. 289-308.