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Adrien Dubouloz

Publications and source records attributed to Adrien Dubouloz.

At least 19 recordsLinked to original sources
arXiv2026

Relative $\mathbb{A}^1$-Contractibility of Smooth Schemes

We study smooth morphisms $f \colon X \to S$ that are $\mathbb{A}^1$-contractible in the unstable $\mathbb{A}^1$-homotopy category $\mathcal{H}(S)$. For base schemes $S$ of finite Krull dimension, we show that $\mathbb{A}^1$-contractibility is a fiberwise property: such a morphism is $\mathbb{A}^1$-contractible if and only if all its geometric fibers are $\mathbb{A}^1$-contractible. We apply this criterion to $\mathbb{A}^n$-fiber spaces, obtaining a geometric description of their $\mathbb{A}^1$-contractibility in terms of local factorizations as towers of torsors under vector bundles, building on results of Asanuma. In low relative dimensions, we establish rigidity results. In relative dimension $1$, $\mathbb{A}^1$-contractible morphisms over normal bases are precisely Zariski locally trivial $\mathbb{A}^1$-bundles. In relative dimension $2$, we show that over bases with characteristic zero residue fields, $\mathbb{A}^1$-contractible morphisms are $\mathbb{A}^2$-fiber spaces, and we obtain Zariski local triviality under additional hypotheses on the base. We also exhibit counterexamples in positive and mixed characteristic and formulate open problems concerning the existence of exotic $\mathbb{A}^1$-contractible surfaces.

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arXiv2026

Cylinders in weighted Fano varieties

Cylinders in Fano varieties receives a lot of attentions recently from the viewpoints of birational geometry and unipotent geometry. In this article, we provide a survey of several known et new results concerning the anti-canonically polar cylindricity of quasi-smooth, well-formed weighted Fano complete intersections in weighted projective spaces.

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arXiv2026

Topologically integrable derivations and additive group actions on affine ind-schemes

We develop a theory of additive group actions on affine ind-schemes through a purely algebraic and topological framework. Affine ind-schemes are described via complete, second-countable, linearly topologized rings, and actions of the additive group are encoded by restricted exponential homomorphisms. We introduce the notion of a topologically integrable derivation, a continuous derivation whose formal exponential converges in the sense of restricted power series, and show that this notion provides the correct extension of locally nilpotent derivations to the infinite-dimensional setting. Our first main result establishes a one-to-one correspondence between topologically integrable derivations and additive group actions on affine ind-schemes, extending the classical correspondence for affine varieties. We then investigate the structure of such actions admitting a slice. In this context, we prove an ind-scheme analog of the classical slice theorem: if an additive group action admits a slice, then the underlying affine ind-scheme is equivariantly isomorphic to a product with the affine line, and the action is given by translation on the second factor. Several examples illustrate the necessity of the topological hypotheses and highlight phenomena absent in the finite-type case.

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arXiv2025

Punctured tubular neighborhoods and stable homotopy at infinity

In this revised version (August 2025), we add a survey of \infty-categorical (co)limits and a replacement lemma for higher functoriality (Lem. 1.4.5), a framework for explicit models of punctured tubular neighborhoods (§3.4), and a new theory of orientation classes for line bundles and Thom spaces of virtual bundles over singular curves (§5.1, 5.2). Building on this, we make explicit the normalization of the resulting isomorphisms, reformulate our main theorem (Th. 5.3.3) to incorporate orientation classes, and show how these choices yield quadratic Mumford matrices computed via Smith form over the Grothendieck-Witt ring of the base field. The appendices are expanded to give a better account of the notion of orientation classes, and to describe trace computations on Chow-Witt groups over possibly non-perfect fields. We warmly thank the referee for insightful comments that motivated us to make our approach much more precise and comprehensive.

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arXiv2025

Algebraic families of higher dimensional $\mathbb{A}^{1}$-contractible affine varieties non-isomorphic to affine spaces

We construct algebraic families of smooth affine $\mathbb{A}^1$-contractible varieties of every dimension $n\geq 4$ over fields of characteristic zero which are non-isomorphic to affine spaces and potential counterexamples to the Zariski Cancellation Problem. We further prove that these families of varieties are also counter examples to the generalized Cancellation problem.

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arXiv2025

The Rigid Pham-Brieskorn Threefolds

We show that a $3$-dimensional Pham-Brieskorn hypersurface $\{ X_0^{a_0} + X_1^{a_1} + X_2^{a_2} + X_3^{a_3}=0\}$ in $\mathbb{A}^4$ such that $\min\{a_0, a_1, a_2, a_3 \} \geq 2$ and at most one element $i$ of $\{0,1,2,3\}$ satisfies $a_i = 2$ does not admit a non-trivial action of the additive group $\mathbb{G}_a$.

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arXiv2024

A polyptych of multi-centered deformation spaces

Extending Verdier's deformation space to the normal cone of a closed subscheme and Rost's double deformation space of a pair of nested closed subschemes, we introduce a notion of deformation spaces attached to chains of immersions of arbitrary lengths $n$. One main result, which builds on the formalism of multi-centered dilatations of schemes, is the existence of so-called panelization isomorphisms, which produce under suitable regularity conditions several canonical isomorphisms between a given deformation space of length $n$ and some deformation spaces of smaller lengths. Having these panelization isomorphisms also allows to give geometric descriptions of the strata -- certain restrictions of special interest -- of deformation spaces. \tableofcontents

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arXiv2024

A survey on algebraic dilatations

In this text, we wish to provide the reader with a short guide to recent works on the theory of dilatations in Commutative Algebra and Algebraic Geometry. These works fall naturally into two categories: one emphasises foundational and theoretical aspects and the other applications to existing theories.

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arXiv2024

Completions of the affine $3$-space into del Pezzo fibrations

We give constructions of completions of the affine $3$-space into total spaces of del Pezzo fibrations of every degree other than $7$ over the projective line. We show in particular that every del Pezzo surface other than $\mathbb{P}^{2}$ blown-up in one or two points can appear as a closed fiber of a del Pezzo fibration $π:X\to\mathbb{P}^{1}$ whose total space $X$ is a $\mathbb{Q}$-factorial threefold with terminal singularities which contains $\mathbb{A}^{3}$ as the complement of the union of a closed fiber of $π$ and a prime divisor $B_{h}$ horizontal for $π$. For such completions, we also give a complete description of integral curves that can appear as general fibers of the induced morphism $\barπ:B_{h}\to\mathbb{P}^{1}$.

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arXiv2023

Relative forms of real algebraic varieties and examples of quasi-projective surfaces with algebraic moduli of real forms

We propose a framework to give a precise meaning to the intuitive notion of "family of real forms of a variety parametrised by a variety" and study some fundamental properties of this notion. As an illustration, for any $n \geq 1$, we construct the first example of a quasi-projective real surface whose mutually non-isomorphic real forms admit a moduli of dimension at least $n$, parametrised by the real points of an affine $n$-space. Expanding on these constructions, we can give quasi-projective real varieties of any dimension whose algebraic moduli of the non-isomorphic real forms has arbitrarily positive dimension.

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arXiv2023

Toric G-solid Fano threefolds

We study toric G-solid Fano threefolds that have at most terminal singularities, where G is an algebraic subgroup of the normalizer of a maximal torus in their automorphism groups.

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arXiv2022

Geometric models for algebraic suspensions

We analyze the question of which motivic homotopy types admit smooth schemes as representatives. We show that given a pointed smooth affine scheme $X$ and an embedding into affine space, the affine deformation space of the embedding gives a model for the ${\mathbb P}^1$ suspension of $X$; we also analyze a host of variations on this observation. Our approach yields many examples of ${\mathbb A}^1$-$(n-1)$-connected smooth affine $2n$-folds and strictly quasi-affine ${\mathbb A}^1$-contractible smooth schemes.

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arXiv2022

Del Pezzo quintics as equivariant compactifications of vector groups

We study faithful actions with a dense orbit of abelian unipotent groups on quintic del Pezzo varieties over a field of characteristic zero. Such varieties are forms of linear sections of the Grassmannian of planes in a 5-dimensional vector space. We characterize which smooth forms admit these types of actions and show that in case of existence, the action is unique up to equivalence by automorphisms. We also give a similar classification for mildly singular quintic del Pezzo threefolds and surfaces.

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arXiv2022

Real frontiers of fake planes

In Dubouloz and Mangolte (Fake real planes: exotic affine algebraic models of $\mathbb{R}^{2}$, arXiv:1507.01574, 2015) 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 $κ\in \{-\infty,0,1,2\}$ which are birationally diffeomorphic to $\mathbb{R}^{2}$.

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arXiv2022

Fake Real Planes: exotic affine algebraic models of $\mathbb{R}^2$

We study real rational models of the euclidean 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, there is only one model. A fake real plane is a nonsingular affine surface defined over the reals with homologically trivial complex locus and real locus diffeomorphic to $\mathbb{R}^2$ but which is not isomorphic to the real affine plane. We prove that fake planes exist by giving many examples and we tackle the question: does there exist fake planes whose real locus is not birationally diffeomorphic to the real affine plane?

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arXiv2022

Fibrations by affine lines on rational affine surfaces with irreducible boundaries

We consider fibrations by affine lines on smooth affine surfaces obtained as complements of smooth rational curves $B$ in smooth projective surfaces $X$ defined over an algebraically closed field of characteristic zero. We observe that except for two exceptions, these surfaces $X \setminus B$ admit infinitely many families of $\mathbb{A}^1$-fibrations over the projective line with irreducible fibers and a unique singular fiber of arbitrarily large multiplicity. For $\mathbb{A}^1$-fibrations over the affine line, we give a new and essentially self-contained proof that the set of equivalence classes of such fibrations up to composition by automorphisms at the source and target is finite if and only if the self-intersection number of $B$ in $X$ is less than or equal to 6.

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