§2 · Agent 2 · Integral models · 2026-10-18T03:21:29Z

Lemma 2.7 · Orbit closures over \(\mathcal O\) have the expected special fibres

proved audit: minor defect

Minor defect (second audit). Re-derived items 1–3: Z_x is B-stable (B×Z_x is flat, hence the closure of its generic fibre) and contains the section y_x; τ̃⁻¹(B) is closed with underlying set |Y_{w₀}| (every geometric point lies in some Y_v by Proposition 2.3, τ̃(Y_v) lies in the cell of v·w₀, which is B iff v=w₀), and an immersion with closed image is a closed immersion; the geometric-fibre induction through P_α·closure(Y_{x′,k}) gives exactly the recursion of Computation 2.6 item 6. Defect that changes no claim: in step 4 'P_αZ_{x′} is closed' in X_O is stated without argument (step 3 proves the analogue only on geometric fibres); it follows in one line from P_α×^B X_O≅ℙ¹_O×X_O and properness of ℙ¹_O over O, with special fibre P_α(Z_{x′}∩X_κ). Supplied in Audit 6.6 step 4, which I checked.

The closure in \(X_{\mathcal O}\) of each generic orbit has special fibre the closure of the special orbit, and closure relations are the same sets \(\Sigma_x\) on every geometric fibre. Induction along type-U raisings from the closed orbit \(x=w_0\), cut out by \(\tilde\tau^{-1}(B)\).

Lemma 2.7.

\(Z_x\) := closure of \(Y_{x,F}\) in \(X_{\mathcal O}\) (reduced, flat, \(B\)-stable); \(\Sigma_x\) as in Computation 2.6, 6, for a fixed choice.

  1. \(\overline{Y_{x,k}}=\bigcup_{v\in\Sigma_x}Y_{v,k}\) for every algebraically closed \(k\) over \(\mathcal O\).
  2. \(Y_{w_0}\) is closed in \(X_{\mathcal O}\).
  3. \(Z_x\cap X_\kappa=\overline{Y_{x,\kappa}}\).
  1. \(Z_x\) is \(B\)-stable (\(B\times Z_x\) flat) and contains the section \(y_x\), so \(Y_{x,\kappa}\subseteq Z_x\).
  2. Item 2. \(\tilde\tau\) maps \(Y_{v,k}\) into the cell \(B\dot{(vw_0)}B\) (Proposition 2.3, step 4), which is \(B\) iff \(v=w_0\); so the closed \(\tilde\tau^{-1}(B)\) has underlying set \(|Y_{w_0}|\).
  3. Item 1 (induction on \(\dim Y_x\)). Base \(x=w_0\): step 2. Else take \(\alpha\) as in Computation 2.6, 3, \(x'=s_\alpha xs_\alpha\): \(P_\alpha\overline{Y_{x',k}}\) is closed (proper map from \(P_\alpha\times^B\overline{Y_{x',k}}\)), irreducible, of dimension \(\dim Y_x\), contains \(Y_{x,k}\), so it is \(\overline{Y_{x,k}}\); by Proposition 2.5, \(P_\alpha Y_v=Y_v\) (\(\alpha\in\Phi_v\)) or \(Y_v\sqcup Y_{s_\alpha vs_\alpha}\), giving \(\Sigma_x\).
  4. Item 3 (same induction). Base: \(Z_{w_0}\subseteq|Y_{w_0}|\) by step 2. Step: \(P_\alpha Z_{x'}\) is closed and contains \(Y_{x,F}\), hence \(Z_x\); its special fibre is \(P_\alpha\overline{Y_{x',\kappa}}=\overline{Y_{x,\kappa}}\). Equality by step 1.
  5. \(\Sigma_v\subseteq\Sigma_x\) for \(v\in\Sigma_x\); item 1 holds with the same \(\Sigma_x\) over \(\overline F\) and \(\overline\kappa\), whatever their characteristics (\(\ne2\)).

Builds on

  • Proposition 2.3 · Borel orbits of \(E_6/F_4\) over an algebraically closed field of characteristic \(\ne2\)
  • Lemma 2.4 · Smooth stabilizers; orbit immersions \(B/B_x\hookrightarrow X\) over \(\mathbb Z[1/2]\)
  • Proposition 2.5 · Rank-one quotients: a point (type \(G\)) or \(\mathbb P^1_{\mathcal O}\) (type \(U\))
  • Computation 2.6 · The \(45\times6\) pairs: 90 of type \(G\), 180 of type \(U\)

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