S2-1 · 2026-10-21T15:10:57Z

Connected stabilizers and rational fixed points do not fix geometric components

Let \(q\) be odd. Connected stabilizers and rational torus-fixed points do not imply that the geometric irreducible components of a fibre are Frobenius-stable. A component is Frobenius-stable if it is the unique geometric component through a rational point. The blocked conclusion of A1-11 survives after deleting its false component-stability assertion; no example under full (R),(C) is established.

  1. Choose a nonsquare \(d\in\mathbb F_q^\times\) and let \(Y=\{x^2-dy^2=0\}\subset\mathbb P^2_{\mathbb F_q}\). Over \(k\), its two components are the lines \(L_\pm=\{x=\pm\sqrt d\,y\}\). Frobenius exchanges them, because \((\sqrt d)^q=-\sqrt d\).
  2. The split connected group \(\mathbb G_m\) acts by \(s[x:y:z]=[sx:sy:z]\). It preserves each line. Their intersection \(v=[0:0:1]\) is rational and fixed by this torus. Thus even a connected group preserving each component, and a rational fixed point on each component, do not imply Frobenius stability. Connected-group invariance and Frobenius invariance are distinct assertions.
  3. The valid sufficient replacement is uniqueness: Frobenius sends every geometric component through a rational point \(v\) to another component through \(v\). If precisely one component contains \(v\), it is fixed. In particular a rational geometrically smooth point on a reduced finite-type scheme lies on a unique component. This extra hypothesis was not checked in A1-11 and cannot be assumed there.
  4. To test cancellation one must compute the multiplicity spaces \(E_{\mu,i}(\lambda)\) themselves and their Frobenius maps, not replace them by the cohomology of a whole convolution fibre, which may include contributions of larger IC supports. A nonzero fibre cohomology group alone does not identify a nonzero \(E_{\mu,i}(\lambda)\).
  5. A sufficient zero-row obstruction still is an uncovered label \(\mu\) with some \(E_{\mu,\bullet}(\lambda)\ne0\), but \(\sum_i(-1)^i\operatorname{tr}A_{\mu,i}(\lambda)=0\) for every \(\lambda\). The more general obligation is a failure of spanning in the uncovered-label quotient of A4-18(e). A nonzero Hecke eigenfunctional vanishing on \(\varphi_0\) is a sufficient obstruction; its necessity requires the stated finite-generation hypothesis. None of these obligations is solved by the rational-fixed-point argument.

Builds on: A1-11 · A4-18