Irreducible Affine Research Articles

Hilbert’s irreducibility theorem for linear differential operators

We prove a differential analogue of Hilbert’s irreducibility theorem. Let L be a linear differential operator with coefficients in C ( X ) ( x ) that is irreducible over C ( X ) ¯ ( x ) , where X is an irreducible affine algebraic variety over an algebraically closed field C of characteristic zero. We show that the set of c ∈ X ( C ) such that the specialized operator L c of L remains irreducible over C ( x ) is Zariski dense in X ( C ) .

Bender–Knuth Billiards in Coxeter Groups

Abstract Let $(W,S)$ be a Coxeter system, and write $S=\{s_i:i\in I\}$ , where I is a finite index set. Fix a nonempty convex subset $\mathscr {L}$ of W. If W is of type A, then $\mathscr {L}$ is the set of linear extensions of a poset, and there are important Bender–Knuth involutions $\mathrm {BK}_i\colon \mathscr {L}\to \mathscr {L}$ indexed by elements of I. For arbitrary W and for each $i\in I$ , we introduce an operator $\tau _i\colon W\to W$ (depending on $\mathscr {L}$ ) that we call a noninvertible Bender–Knuth toggle; this operator restricts to an involution on $\mathscr {L}$ that coincides with $\mathrm {BK}_i$ in type A. Given a Coxeter element $c=s_{i_n}\cdots s_{i_1}$ , we consider the operator $\mathrm {Pro}_c=\tau _{i_n}\cdots \tau _{i_1}$ . We say W is futuristic if for every nonempty finite convex set $\mathscr {L}$ , every Coxeter element c and every $u\in W$ , there exists an integer $K\geq 0$ such that $\mathrm {Pro}_c^K(u)\in \mathscr {L}$ . We prove that finite Coxeter groups, right-angled Coxeter groups, rank-3 Coxeter groups, affine Coxeter groups of types $\widetilde A$ and $\widetilde C$ , and Coxeter groups whose Coxeter graphs are complete are all futuristic. When W is finite, we actually prove that if $s_{i_N}\cdots s_{i_1}$ is a reduced expression for the long element of W, then $\tau _{i_N}\cdots \tau _{i_1}(W)=\mathscr {L}$ ; this allows us to determine the smallest integer $\mathrm {M}(c)$ such that $\mathrm {Pro}_c^{{\mathrm {M}}(c)}(W)=\mathscr {L}$ for all $\mathscr {L}$ . We also exhibit infinitely many non-futuristic Coxeter groups, including all irreducible affine Coxeter groups that are not of type $\widetilde A$ , $\widetilde C$ , or $\widetilde G_2$ .

Coarse cubical rigidity

AbstractWe show that for many right‐angled Artin and Coxeter groups, all cocompact cubulations coarsely look the same: They induce the same coarse median structure on the group. These are the first examples of non‐hyperbolic groups with this property. For all graph products of finite groups and for Coxeter groups with no irreducible affine parabolic subgroups of rank , we show that all automorphisms preserve the coarse median structure induced, respectively, by the Davis complex and the Niblo–Reeves cubulation. As a consequence, automorphisms of these groups have nice fixed subgroups and satisfy Nielsen realisation.

Representations Of Lie Algebras of Vector Fields on Pairing Between Gauge Modules and Rudakov Modules

For an irreducible affine variety X over an algebraically closed field of characteristic zero, we define two new classes of modules over the Lie algebra of vector fields on X: gauge modules and Rudakov modules. These modules admit a compatible action of the algebra of functions on X. We prove general simplicity theorems for these two types of modules, demonstrating their irreducibility under specific conditions. Additionally, we establish a pairing between gauge modules and Rudakov modules, highlighting the connections and interactions between these two classes of modules. We have established that gauge modules and Rudakov modules, corresponding to simple glN -modules, remain irreducible as modules over the Lie algebra of vector fields unless they appear in the de Rham complex. Additionally, we have studied the irreducibility of tensor products of Rudakov modules, providing a comprehensive understanding of these module structures and their applications.

On free and plus-one generated curves arising from free curves by addition–deletion of a line

In a recent paper, after introducing the notion of plus-one generated hyperplane arrangements, Takuro Abe has shown that if we add (respectively, delete) a line to (respectively, from) a free line arrangement, then the resulting line arrangement is either free or plus-one generated. In this paper, we prove that the same properties hold when we replace the line arrangement by a free curve and add (respectively, delete) a line. The proof uses a new version of a key result due originally to H. Schenck, H. Terao and M. Yoshinaga, in which no quasi-homogeneity assumption is needed. Two conjectures about the Tjurina number of a union of two plane curve singularities are also stated. As a geometric application, we show that, under a mild numerical condition, the projective closure of a contractible, irreducible affine plane curve is either free or plus-one generated, using a deep result due to U. Walther.

Dimension Growth for Affine Varieties

Abstract We prove uniform upper bounds on the number of integral points of bounded height on affine varieties. If $X$ is an irreducible affine variety of degree $d\geq 4$ in ${\mathbb{A}}^{n}$, which is not the preimage of a curve under a linear map ${\mathbb{A}}^{n}\to{\mathbb{A}}^{n-\dim X+1}$, then we prove that $X$ has at most $O_{d,n,\varepsilon }(B^{\dim X - 1 + \varepsilon })$ integral points up to height $B$. This is a strong analogue of dimension growth for projective varieties, and improves upon a theorem due to Pila, and a theorem due to Browning–Heath-Brown–Salberger. Our techniques follow the $p$-adic determinant method, in the spirit of Heath-Brown, but with improvements due to Salberger, Walsh, and Castryck–Cluckers–Dittmann–Nguyen. The main difficulty is to count integral points on lines on an affine surface in ${\mathbb{A}}^{3}$, for which we develop point-counting results for curves in ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$. We also formulate and prove analogous results over global fields, following work by Paredes–Sasyk.

On quotients of Coxeter groups

On quotients of Coxeter groups

Isometries of wall-connected twin buildings

Abstract We introduce the notion of a wall-connected twin building and show that the local-to-global principle holds for these twin buildings. As each twin building satisfying Condition (co) (introduced in [7]) is wall-connected, we obtain a strengthening of the main result of [7] that covers also the thick irreducible affine twin buildings of rank at least 3.

Extended Affine Root Supersystems of Types $C(I, J)$ and $BC(1, 1)$

In this paper, we complete the characterization of tame irreducible extended affine root supersystems. We give a complete description of tame irreducible extended affine root supersystems of type X = C(1, 1),C(1, 2), C(2, 2) and BC(1, 1) and determine isomorphic classes.

The second lowest two-sided cell in the affine Weyl group [formula omitted

The second lowest two-sided cell in the affine Weyl group [formula omitted

Sums of even powers of [formula omitted]-regulous functions

Sums of even powers of [formula omitted]-regulous functions

Minimal Rational Curves and 1-Flat Irreducible G-Structures

1-Flat irreducible G-structures, equivalently, irreducible G-structures admitting torsion-free affine connections, have been studied extensively in differential geometry, especially in connection with the theory of affine holonomy groups. We propose to study them in a setting in algebraic geometry, where they arise from varieties of minimal rational tangents (VMRT) associated to families of minimal rational curves on uniruled projective manifolds. We prove that such a structure is locally symmetric when the dimension of the uniruled projective manifold is at least 5. By the classification result of Merkulov and Schwachhöfer on irreducible affine holonomy, the problem is reduced to the case when the VMRT at a general point of the uniruled projective manifold is isomorphic to a subadjoint variety. In the latter situation, we prove a stronger result that, without the assumption of 1-flatness, the structure arising from VMRT is always locally flat. The proof employs the method of Cartan connections. An interesting feature is that Cartan connections are considered not for the G-structures themselves, but for certain geometric structures on the spaces of minimal rational curves.

On the Existence of B-Root Subgroups on Affine Spherical Varieties

Let X be an irreducible affine algebraic variety that is spherical with respect to an action of a connected reductive group G. In this paper, we provide sufficient conditions, formulated in terms of weight combinatorics, for the existence of one-parameter additive actions on $$X$$ normalized by a Borel subgroup $$B \subset G$$ . As an application, we prove that every G-stable prime divisor in X can be connected with an open G-orbit by means of a suitable B-normalized one-parameter additive action.

CHARACTERIZATION OF n-DIMENSIONAL NORMAL AFFINE SLn-VARIETIES

We show that any normal irreducible affine n-dimensional SLn-variety X is determined by its automorphism group seen as an ind-group in the category of normal irreducible affine varieties. In other words, if Y is an irreducible affine normal algebraic variety such that Aut(Y) ≃ Aut(X) as an ind-group, then Y ≃ X as a variety. If we drop the condition of normality on Y , then this statement fails. In case n ≥ 3, the result above holds true if we replace Aut(X) by \U0001d4b0(X), where \U0001d4b0(X) is the subgroup of Aut(X) generated by all one-dimensional unipotent subgroups. In dimension 2 we have some interesting exceptions.

SIMPLICITY CRITERIA FOR RINGS OF DIFFERENTIAL OPERATORS

Abstract Let K be a field of arbitrary characteristic, $${\cal A}$$ be a commutative K-algebra which is a domain of essentially finite type (e.g., the algebra of functions on an irreducible affine algebraic variety), $${a_r}$$ be its Jacobian ideal, and $${\cal D}\left( {\cal A} \right)$$ be the algebra of differential operators on the algebra $${\cal A}$$ . The aim of the paper is to give a simplicity criterion for the algebra $${\cal D}\left( {\cal A} \right)$$ : the algebra $${\cal D}\left( {\cal A} \right)$$ is simple iff $${\cal D}\left( {\cal A} \right)a_r^i{\cal D}\left( {\cal A} \right) = {\cal D}\left( {\cal A} \right)$$ for all i ≥ 1 provided the field K is a perfect field. Furthermore, a simplicity criterion is given for the algebra $${\cal D}\left( R \right)$$ of differential operators on an arbitrary commutative algebra R over an arbitrary field. This gives an answer to an old question to find a simplicity criterion for algebras of differential operators.

The Wadge hierarchy on Zariski topologies

The Wadge hierarchy on Zariski topologies

Windows for cdgas

Windows for cdgas

A strongly irreducible affine iterated function system with two invariant measures of maximal dimension

AbstractA classical theorem of Hutchinson asserts that if an iterated function system acts on $\mathbb {R}^{d}$ by similitudes and satisfies the open set condition then it admits a unique self-similar measure with Hausdorff dimension equal to the dimension of the attractor. In the class of measures on the attractor, which arise as the projections of shift-invariant measures on the coding space, this self-similar measure is the unique measure of maximal dimension. In the context of affine iterated function systems it is known that there may be multiple shift-invariant measures of maximal dimension if the linear parts of the affinities share a common invariant subspace, or more generally if they preserve a finite union of proper subspaces of $\mathbb {R}^{d}$ . In this paper we give an example where multiple invariant measures of maximal dimension exist even though the linear parts of the affinities do not preserve a finite union of proper subspaces.

Difference Galois groups under specialization

We present a difference analogue of a result given by Hrushovski on differential Galois groups under specialization. Let k k be an algebraically closed field of characteristic zero and let X \mathbb {X} be an irreducible affine algebraic variety over k k . Consider the linear difference equation σ ( Y ) = A Y , \begin{equation*} \sigma (Y)=AY, \end{equation*} where A ∈ G L n ( k ( X ) ( x ) ) A\in \mathrm {GL}_n(k(\mathbb {X})(x)) and σ \sigma is the shift operator σ ( x ) = x + 1 \sigma (x)=x+1 . Assume that the Galois group G G of the above equation over k ( X ) ¯ ( x ) \overline {k(\mathbb {X})}(x) is defined over k ( X ) k(\mathbb {X}) , i.e., the vanishing ideal of G G is generated by a finite set S ⊂ k ( X ) [ X , 1 / det ( X ) ] S\subset k(\mathbb {X})[X,1/\det (X)] . For a c ∈ X {\mathbf {c}}\in \mathbb {X} , denote by v c v_{{\mathbf {c}}} the map from k [ X ] k[\mathbb {X}] to k k given by v c ( f ) = f ( c ) v_{{\mathbf {c}}}(f)=f({\mathbf {c}}) for any f ∈ k [ X ] f\in k[\mathbb {X}] . We prove that the set of c ∈ X {\mathbf {c}}\in \mathbb {X} satisfying that v c ( A ) v_{\mathbf {c}}(A) and v c ( S ) v_{\mathbf {c}}(S) are well-defined and the affine variety in G L n ( k ) \mathrm {GL}_n(k) defined by v c ( S ) v_{{\mathbf {c}}}(S) is the Galois group of σ ( Y ) = v c ( A ) Y \sigma (Y)=v_{{\mathbf {c}}}(A)Y over k ( x ) k(x) is Zariski dense in X \mathbb {X} . We apply our result to van der Put-Singer’s conjecture which asserts that an algebraic subgroup G G of G L n ( k ) \mathrm {GL}_n(k) is the Galois group of a linear difference equation over k ( x ) k(x) if and only if the quotient G / G ∘ G/G^\circ by the identity component is cyclic. We show that if van der Put-Singer’s conjecture is true for k = C k=\mathbb {C} , then it will be true for any algebraically closed field k k of characteristic zero.

On the preimage of a sphere by a polynomial mapping

Let X be an irreducible complex affine variety of dimension greater than one and let f:X rightarrow mathbb {C}^m be a polynomial mapping. Let {|}*{|} be a semialgebraic norm on mathbb {C}^m. Then for R large enough the sets f^{-1}(B_R), f^{-1}(S_R), X{setminus } f^{-1}(B_R) are all connected, where B_R={ zin mathbb {C}^m : |z|le R} and S_R={ zin mathbb {C}^m : |z| = R}. As an application we show that if F is a counterexample to the Jacobian Conjecture, then the non-properness set of F has a non-trivial link at infinity.