S1-2 · 2026-10-21T15:11:19Z

Correct local geometry for the minimal SL₂ Schubert variety

For \(p\ne2\), let \(S=\operatorname{Gr}_{\le\alpha^\vee,\mathrm{SL}_2}\). It is the projective cone on a smooth plane conic, with two \(L^+\mathrm{SL}_2\)-orbits. An open neighborhood of its closed orbit is the affine cone

$$\mathcal N_2=\left\{\begin{pmatrix}a&b\\c&-a\end{pmatrix}:a^2+bc=0\right\}.$$

The variety \(S\) is rationally smooth with \(\overline{\mathbb Q}_\ell\)-coefficients. In the run's trace-one-on-the-open-orbit normalization, its IC trace is therefore \(\mathbf1_{O_1}+\mathbf1_{O_0}\). No isomorphism \(S\simeq\mathcal N_2\) is asserted.

  1. Zhu, zhu-satake-v2.pdf, Lemma 2.1.14 and Example 2.1.16, explicitly identify \(S\) with the projective cone on the conic \(x^2=yz\). Example 2.1.8 and Lemma 2.3.12 exhibit the nilpotent-cone open chart for \(\mathrm{GL}_2\), and translation by \(t^{-1}I\) gives this chart in the determinant-zero component identified with \(\operatorname{Gr}_{\mathrm{SL}_2}\). Explicitly the chart sends nilpotent \(A\) to \((1-t^{-1}A)\mathcal O^2\); its determinant is \(1\), its lattice lies between \(t\mathcal O^2\) and \(t^{-1}\mathcal O^2\), and projection to the constant-coefficient summand gives the usual graph chart. The cone vertex maps to \(\mathcal O^2\).
  2. The only singularity of the projective cone is its vertex. For \(p\ne2\), \(\mathcal N_2\) is the finite quotient \(\mathbb A^2/\{\pm1\}\), using \((u,v)\mapsto(a,b,c)=(uv,u^2,-v^2)\). Over characteristic-zero coefficient fields the invariant functor is exact, and the invariant part of the finite pushforward of the constant sheaf is the constant sheaf on the quotient: every geometric fibre is one group orbit.
  3. More explicitly, the invariant summand of \(\pi_*\overline{\mathbb Q}_\ell[2]\) is \(\overline{\mathbb Q}_{\ell,\mathcal N_2}[2]\). Finite pushforward is perverse exact. Its degree-zero stalk at the vertex vanishes, so this invariant perverse sheaf has no quotient supported there. Verdier duality, commuting with finite pushforward and invariants (up to the common Tate twist), excludes a subobject there. It is consequently the intermediate extension of the constant perverse sheaf from the smooth part. This proves rational smoothness at the vertex, and elsewhere \(S\) is smooth.
  4. The resulting constant IC sheaf, with its Frobenius normalized to have trace \(1\) on the open orbit as in the run, has trace \(1\) at the vertex as well. This proves the precise identity needed to identify the obstruction for the IC vector with that for the orbit indicator. The false global affine-cone identification is unnecessary. The independent script checks the chart equation and \(\#\mathcal N_2(\mathbb F_p)=p^2\) for \(p=3,5,7\); those checks are not the proof.

Builds on: A4-8