The purpose of this paper is to show that, under mild inductive assumptions, if a group $G$ contains a component that is a simple group of Lie type in characteristic $p$ and $O_p(G) = 1$, then the Quillen poset of $G$ at $p$ has nonzero rational homology. In particular, this shows that such components cannot arise in a minimal counterexample to Quillen's conjecture. This result is of particular interest at the prime $p=2$, where the conjecture is still open.
In these notes, we introduce the 50-year-old $K(\pi, 1)$ conjecture alongside Coxeter and Artin groups. Roughly speaking, the conjecture states that the complement in $\mathbb{C}^n$ of a"symmetric"configuration of hyperplanes is a $K(\pi, 1)$ space. Our end goal is to present a proof of the conjecture in the so-called spherical case, where only a finite number of hyperplanes are removed, through methods from combinatorial topology. This proof draws inspiration from the original proof of the spherical case, which is a special case of a celebrated 1972 theorem by Pierre Deligne.
Giovanni Paolini· Winter Braids Lecture Notes· 0 citations
Let $p, N \geq 5$ be primes such that $N \equiv 1 \bmod p$. We prove modularity theorems at levels $N$ and $N^2$, showing that suitable Eisenstein localizations of the weight-$2$ $p$-adic Hecke algebra at these levels are isomorphic to certain natural quotients of a universal pseudodeformation ring. This universal ring parametrizes pseudorepresentations that are residually Eisenstein, unramified outside $Np$ and finite-flat at $p$, and satisfy appropriate conditions at $N$ depending on the level.
Let $(p,q)$ be a Jacobian pair. We show that the real Jacobian conjecture holds if the degree of $p$ is 7 and the highest degree homogenous part is of the form $\alpha x^7 + \beta x^6y$ for $\alpha^2+\beta^2 \neq 0$. We then show that there are no atypical Jacobian pairs such that $\mathrm{deg} \, p =7$ and $\mathrm{deg}\, q$ is even and coprime with 7.
Assuming the log resolution conjecture for all log pairs birational to $X$, we prove the cone theorem for projective log canonical, $\mathbb{Q}$-factorial fourfold pairs $(X,\Delta)$ such that $K_X +\Delta\equiv M \geq 0$ over bases of positive and mixed characteristic $p>5$.
Let $k$ be a field of characteristic $2$. We exhibit an explicit polynomial endomorphism $F: \mathbb{A}_k^3\to\mathbb{A}_k^3$ whose Jacobian determinant is identically $1$, whose induced extension of rational function fields has degree $3$, and which is nevertheless noninjective. Since $2\neq 3$, this gives a counterexample to the usual Adjamagbo, or separable, formulation of the Jacobian conjecture in characteristic $2$. Stabilization yields analogous counterexamples in every dimension $n\geq 3$
Let $X$ be an algebraic variety of dimension $d$ over an algebraically closed field $k$ of zero characteristic. Suppose that $G$ is a virtually polycyclic subgroup in the group $\mathrm{Bir}(X)$ of birational automorphisms of $X$. We show that the virtual derived length of $G$ does not exceed $2d+1$. Moreover, if $X$ is a surface, the bound can be improved to $3$, and this value is optimal.
A. Golota· 0 citations
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