Summary: Let \(G\) be a connected semi-simple algebraic group defined over an algebraically closed field \(k\), and let \(T\subset B \subset P\) be respectively a maximal torus, a Borel subgroup and a parabolic subgroup of \(G\). Inspired by a beautiful result of Dale Peterson (unpublished) describing the singular locus of a Schubert variety in \(G/B\), we characterize the \(T\)-fixed points in the singular locus of an arbitrary irreducible \(T\)-stable subvariety of \(G/P\) (a \(T\)-variety for short).
Peterson’s result (cf. §1) says that if \(k=\mathbb C\), then a Schubert variety \(X \subset G/B\) is smooth at a \(T\)-fixed point \(x\) if and only if it is smooth at every \(T\)-fixed point \(y>x\) (in the Bruhat-Chevalley order on the fixed point set \(X^T\)) and all the limits \(\tau_C(X,x) = \lim_{z\to x} T_z(X)\) (\(z\in C\setminus C^T\)) of the Zariski tangent spaces \(T_z(X)\) of \(X\) coincide as \(C\) varies over the set of all \(T\)-stable curves in \(X\) with \(C^T=\{x,y\}\), where \(y>x\). Using this, Peterson showed that if \(G\) is simply laced (and defined over \(\mathbb C\)), then every rationally smooth point of a Schubert variety in \(G/B\) is smooth. More generally, the deformation \(\tau_C(X,x)\) is defined for any \(k\)-variety \(X\) with a \(T\)-action provided \(C\) is what we call good, i.e. \(C\) is a curve of the form \(C=\overline{Tz}\), where \(z\) is a smooth point of \(X\setminus X^T\) and \(x\in C^T\).
Our first main result (theorem 1.4) says that if \(x\in X\) is an attractive fixed point, then \(X\) is smooth at \(x\) if and only if there exist at least two good \(C\) containing \(x\) such that \(\tau_C(X,x) = \text{TE}(X,x)\), where \(\text{TE}(X,x)\) denotes the span of the tangent lines of the \(T\)-stable curves in \(X\) containing \(x\). In addition, if \(X\) is Cohen-Macaulay at \(x\) and \(\tau_C(X,x)= \text{TE}(X,x)\) for even one good \(C\), then \(X\) is smooth at \(x\).
Our second main result (theorem 1.6) says that if \(X\) is a \(T\)-variety in \(G/P\), where \(G\) is simply laced, then \(\tau_C(X,x) \subset \text{TE}(X,x)\) for each good \(C\). This is not true for general \(G\), but when \(G\) has no \(G_2\) factors, then \(\tau_C(X,x)\) is always contained in the linear span of the reduced tangent cone to \(X\) at \(x\). These results lead to several descriptions of the smooth fixed points of a \(T\)-variety in \(G/P\) and, in particular, they give simple proofs of Peterson’s results valid for any algebraically closed field.
We also show (cf. example 7.1) that there can exist \(T\)-stable subvarieties in \(G/B\), where \(G\) is simply laced, which have rationally smooth \(T\)-fixed points in their singular loci.