Given a field K, we investigate which subgroups of the group AutAK2 of polynomial automorphisms of the plane are linear or not.
The results are contrasted. The group AutAK2 itself is nonlinear, ...except if K is finite, but it contains some large subgroups, of “codimension-five” or more, which are linear. This phenomenon is specific to dimension two: it is easy to prove that any natural “finite-codimensional” subgroup of AutAK3 is nonlinear, even for a finite field K.
When chK=0, we also look at a similar questions for f.g. subgroups, and the results are again disparate. The group AutAK2 has a one-related f.g. subgroup which is not linear. However, there is a large subgroup, of “codimension-three”, which is locally linear but not linear.
We prove several results concerning automorphism groups of quasismooth complex weighted projective hypersurfaces; these generalize and strengthen existing results for hypersurfaces in ordinary ...projective space. First, we prove in most cases that automorphisms extend to the ambient weighted projective space. We next provide a characterization of when the linear automorphism group is finite and find an explicit uniform upper bound on the size of this group. Finally, we describe the automorphisms of a generic quasismooth hypersurface with given weights and degree.
Let X be a (projective, geometrically irreducible, non-singular) algebraic curve of genus g≥2 defined over an algebraically closed field K of odd characteristic p. Let Aut(X) be the group of all ...automorphisms of X which fix K element-wise. It is known that if |Aut(X)|≥8g3 then the p-rank (equivalently, the Hasse–Witt invariant) of X is zero. This raises the problem of determining the (minimum-value) function f(g) such that whenever |Aut(X)|≥f(g) then X has zero p-rank. For eveng we prove that f(g)≤900g2. The odd genus case appears to be much more difficult although, for any genus g≥2, if Aut(X) has a solvable subgroup G such that |G|>252g2 then X has zero p-rank and G fixes a point of X. Our proofs use the Hurwitz genus formula and the Deuring Shafarevich formula together with a few deep results from Group theory characterizing finite simple groups whose Sylow 2-subgroups have a cyclic subgroup of index 2. We also point out some connections with the Abhyankar conjecture and the Katz–Gabber covers.
For each finite primitive subgroup G of PGL4(C), we find all the smooth G-invariant quartic surfaces. We also find all the faithful representations in PGL4(C) of the smooth quartic G-invariant ...surfaces by the groups: A5,S5,PSL2(F7), A6, Z24⋊Z5 and Z24⋊D10. The primitive representation of these groups is precisely the subgroups of PGL4(C) for which P3 is not G-super rigid. As a byproduct, we show that the smooth quartic surface with the biggest group of projective automorphism is given by {x04+x14+x24+x34+12x0x1x2x3=0} (unique up to projective equivalence).
We show that if a subgroup of the automorphism group of the Fraïssé limit of finite Heyting algebras has a countable index, then it lies between the pointwise and setwise stabilizer of some finite ...set.
The minimal degree of a permutation group G is the minimum number of points not fixed by non-identity elements of G. Lower bounds on the minimal degree have strong structural consequences on G. Babai ...conjectured that if a primitive coherent configuration with n vertices is not a Cameron scheme, then its automorphism group has minimal degree ≥cn for some constant c>0. In 2014, Babai proved the desired lower bound on the minimal degree of the automorphism groups of strongly regular graphs, thus confirming the conjecture for primitive coherent configurations of rank 3.
In this paper, we extend Babai's result to primitive coherent configurations of rank 4, confirming the conjecture in this special case. The proofs combine structural and spectral methods.
Recently (March 2022) Sean Eberhard published a class of counterexamples of rank 28 to Babai's conjecture and suggested to replace “Cameron schemes” in the conjecture with a more general class he calls “Cameron sandwiches”. Naturally, our result also confirms the rank 4 case of Eberhard's version of the conjecture.
Let G be a subgroup of the three dimensional projective group PGL(3,q) defined over a finite field Fq of order q, viewed as a subgroup of PGL(3,K) where K is an algebraic closure of Fq. For ...G≅PGL(3,q) and for the seven nonsporadic, maximal subgroups G of PGL(3,q), we investigate the (projective, irreducible) plane curves defined over K that are left invariant by G. For each, we compute the minimum degree d(G) of G-invariant curves, provide a classification of all G-invariant curves of degree d(G), and determine the first gap ε(G) in the spectrum of the degrees of all G-invariant curves. We show that the curves of degree d(G) belong to a pencil depending on G, unless they are uniquely determined by G. For most examples of plane curves left invariant by a large subgroup of PGL(3,q), the whole automorphism group of the curve is linear, i.e., a subgroup of PGL(3,K). Although this appears to be a general behavior, we show that the opposite case can also occur for some irreducible plane curves, that is, the curve has a large group of linear automorphisms, but its full automorphism group is nonlinear.
Katsura and Oort obtained an explicit description of the supersingular locus S3,1 of the Siegel modular variety of degree 3 in terms of class numbers. In this paper we study an alternative ...stratification of S3,1, the so-called mass stratification. We show that when p≠2, there are eleven strata (one of a-number 3, two of a-number 2 and eight of a-number 1). We give an explicit mass formula for each stratum and classify possible automorphism groups on each stratum of a-number one. On the largest open stratum we show that every automorphism group is {±1} if and only if p≠2; that is, we prove that Oort's conjecture on the automorphism groups of generic supersingular abelian threefolds holds precisely when p>2.
We consider a certain subgroup An+ of the automorphism group of a free group of rank n. It can be regarded as a free group analogue of the group Λn of integral lower-triangular matrices. We call An+ ...the lower-triangular automorphism group of a free group. The first aim of the paper is to give a finite presentation for An+.
The abelianization of the free group induces the surjective homomorphism ρ+ from An+ to Λn. In our previous paper 18, we introduced the lower-triangular IA-automorphism group IAn+. Here we show that IAn+ coincides with the kernel of ρ+. The second aim of the paper is to give an infinite presentation for IAn+.
Finally, we study a relation of the second homology groups between An+ and Λn. In particular, we compute the second homology group H2(Λn,L) by using Magnus's presentation where L is a principal ideal domain in which 2 is invertible. For example, L=Q,Z/pZ for prime p≥3. This gives a lower bound on the integral second homology group of An+.
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