Bjorn Poonen
The problem of deciding, given a complex variety X, a point x in X, and a subvariety Z of X, whether there is an automorphism of X mapping x into Z is proved undecidable. Along the way, we prove the undecidability of a version of Hilbert's tenth problem for systems of polynomials over Z defining an affine Q-variety whose projective closure is smooth.
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We prove that a quasi-finite endomorphism of an algebraic variety over an algebraically closed field of characteristic zero, that is injective on the complement of a closed subvariety, is an automorphism. We also prove that an endomorphism of complex algebraic variety that is injective on the complement of a closed subvariety of codimension at least $2$, is an automorphism.
We show that for an irreducible subvariety Y of an abelian variety X the Gauss mapping, from the conormal bundle of $Y$ to the dual of the tangent space of $X$ at the origin, is not dominant if and only if Y is degenerate in the sense that there exists a nontrivial abelian subvariety A of X such that A+Y=Y holds
We study the symplectic leaves of the subvariety of fixed points of an automorphism of a Calogero-Moser space induced by an element of finite order of the normalizer of the associated complex reflection group $W$. We give a parametrization {\it à la Harish-Chandra} of its symplectic leaves (generalizing earlier works of Bellamy and Losev). This result is inspired by the mysterious relations between the geometry of Calogero-Moser spaces and unipotent representations of finite reductive groups, which will be the theme of a forthcoming paper.
We study the subvariety of fixed points of an automorphism of a Calogero-Moser space induced by a regular element of finite order of the normalizer of the associated complex reflection group $W$. We determine some of (and conjecturally all) the ${\mathbb{C}}^\times$-fixed points of its unique irreducible component of maximal dimension in terms of the character table of $W$. This is inspired by the mysterious relations between the geometry of Calogero-Moser spaces and unipotent representations of finite reductive groups, which will be the theme of a forthcoming paper.
We study genus 3 hyperelliptic curves which have an extra involution. The locus $Ł_3$ of these curves is a 3-dimensional subvariety in the genus 3 hyperelliptic moduli $\H_3$. We find a birational parametrization of this locus by affine 3-space. For every moduli point $\p \in \H_3$ such that $|\Aut (\p)|>2$, the field of moduli is a field of definition. We provide a rational model of the curve over its field of moduli for all moduli points $\p \in \H_3$ such that $|\Aut(\p)|>4$. This is the first time that such a rational model of these curves appears in the literature.
We give some results concerning the smoothness of the image of a real-analytic submanifold in complex space under the action of a finite holomorphic mapping. For instance, if the submanifold is not contained in a proper complex subvariety, we give a necessary and sufficient condition guaranteeing that its image is smooth and the mapping is transversal to the image.
An affine variety $X$ of dimension $\ge 2$ is called {\em flexible} if its special automorphism group SAut$(X)$ acts transitively on the smooth locus $X_{reg}$ \cite{AKZ}. Recall that the special automorphism group SAut$(X)$ is the subgroup of the automorphism group Aut$(X)$ generated by all one-parameter unipotent subgroups \cite{AKZ}. Given a normal, flexible, affine variety $X$ and a closed subvariety $Y$ in $X$ of codimension at least 2, we show that the pointwise stabilizer subgroup of $Y$ in the group SAut$(X)$ acts infinitely transitively on the complement $X\backslash Y$, that is, $m$-…
On the automorphism group of a toral variety
本文研究特征零代数闭域上环面簇的正则自同构群，证明其可按极大环面分解为直积，并给出由剩余因子自同构群计算原群及显式描述的条件。
Automorphisms of the procongruence pants complex
该研究证明了扩展映射类群的共轭完成的自同构均为内自同构，并展示了其与拼接裤复形自同构群的同构关系。
Let $k$ be a field of characteristic zero. Let $F = X + H$ be a polynomial mapping from $k^n \to k^n$, where $X$ is the identity mapping and $H$ has only degree two terms and higher. We say that the Jacobian matrix $JH$ of $H$ is strongly nilpotent with index $p$ if for all $X^{(1)},\ldots,X^{(p)} \in k^n$ we have \begin{align*} JH(X^{(1)})\ldots JH (X^{(p)}) = 0. \end{align*} Every $F$ of this form is a polynomial automorphism, i.e. there is a second polynomial mapping $F^{-1}$ such that $F \circ F^{-1} = F^{-1} \circ F = X$. We prove that the degree of the inverse $F^{-1}$ satisfies \begin{a…
Automorphisms of profinite mapping class groups
本文研究了闭合定向可微流形的纯映射类群的 profinite 完成的自同构群，在自然刚性条件下确定了 profinite Grothendieck-Teichmüller 群嵌入到 profinite 完成的外自同构群中的结果。
Let $M\subset C^N$ be a minimal real-analytic CR-submanifold and $M'\subset C^{N'}$ a real-algebraic subset through points $p\in M$ and $p'\in M'$. We show that that any formal (holomorphic) mapping $f\colon (C^N,p)\to (C^{N'},p')$, sending $M$ into $M'$, can be approximated up to any given order at $p$ by a convergent map sending $M$ into $M'$. If $M$ is furthermore generic, we also show that any such map $f$, that is not convergent, must send (in an appropriate sense) $M$ into the set $E'\subset M'$ of points of D'Angelo infinite type. Therefore, if $M'$ does not contain any nontrivial compl…
Let K[x,y] be the algebra of two-variable polynomials over a field K. A polynomial p=p(x, y) is called a test polynomial (for automorphisms) if, whenever ϕ(p)=p for a mapping ϕof K[x,y], this ϕmust be an automorphism. Here we show that p \in C[x,y] is a test polynomial if and only if p does not belong to any proper retract of C[x,y]. This has the following corollary that may have application to the Jacobian conjecture: if a mapping ϕof C[x,y] with invertible Jacobian matrix is ``invertible on one particular polynomial", then it is an automorphism. More formally: if there is a non-constant poly…
The point of this paper is to use affine automorphisms from algebraic geometry to build cryptographic multivariate mappings. We will construct groups G,H, both isomorphic to the cyclic group with a prime number of elements and multilinear pairings from the k-fold product of G to H. The construction is reminiscent of techniques in multivariate encryption. We display several different versions of the discrete logarithm problem for these groups. We show that the efficient solution of some of these problems result in efficient algorithms for inverting systems of multivariate polynomials correspond…
Automorphisms of procongruence curve and pants complexes
本文通过研究procongruence映射类群在曲线和裤子复形上的作用，证明了裤子复形procongruence完备化的刚性定理，并由此得出光滑代数曲线模栈满足弱anabelan性质。
Automorphisms of GKM graphs and regular semisimple Hessenberg varieties
本文研究了正则半单 Hessenberg 簇的自同构群结构，证明了其连通分支的约化部分为代数环面，商群同构于对称群的子群，并得出任意射影 GKM 流形的自同构群商为有限群的结论。
On mappings with Jacobian one
研究次数至多为d且雅可比行列式为1的多项式自同构集合\(A(n, d)\)，证明其为扎里斯基闭集，其不可约分支与雅可比猜想相关，分支维数至少为\(n^2 - 1\)，在特定条件下集合\(X(n,d)\)的一般元素是雅可比猜想反例。
Let A^2 be the affine plane over a field K of characteristic 0. Birational morphisms of A^2 are mappings A^2 \to A^2 given by polynomial mappings ϕof the polynomial algebra K[x,y] such that for the quotient fields, one has K(ϕ(x), ϕ(y)) = K(x,y). Polynomial automorphisms are obvious examples of such mappings. Another obvious example is the mapping τ_x given by x \to x, y \to xy. For a while, it was an open question whether every birational morphism is a product of polynomial automorphisms and copies of τ_x. This question was answered in the negative by P. Russell (in an informal communication)…
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