How singular is a Schubert variety?
A permutation corresponds to a Schubert variety, which is a space of flags. Here we illustrate six conditions on it, from smooth to Cohen–Macaulay: five measure how mild its singularities are, and one, being defined by inclusions, is about the equations that cut it out. We show what each one says geometrically, what the simplest variety that fails it looks like, and how to read it off the permutation.
From a permutation to a variety
A flag in \(\mathbb{C}^n\) is a chain of subspaces \(F_1 \subset F_2 \subset \cdots \subset F_{n-1}\) with \(\dim F_p = p\). Fix the reference flag \(E\), in which \(E_q\) is spanned by the first \(q\) standard basis vectors. For a permutation \(w\) of \(1, \ldots, n\), the Schubert variety \(X_w\) is the set of flags \(F\) such that \(\dim(F_p \cap E_q) \ge r_w(p, q)\) for all \(p\) and \(q\), where \(r_w(p, q)\) is the number of points of \(w\) among its first \(p\) positions with values at most \(q\). The dimension of \(X_w\) is \(\ell(w)\), the number of inversions of \(w\).
In \(\mathbb{C}^3\) a flag is a line through the origin in a plane. In \(\mathbb{C}^4\) a flag is a line through the origin in a plane in a three-dimensional space. In projective three-dimensional space these become a point on a line in a plane, and that is how we draw them. The picture of length 3 shows the same step one dimension down, on a square in the plane \(x = 1\). The reference flag is gray. The flag in color moves through \(X_w\), so that at every moment it is a flag of \(X_w\). Beside each picture, the conditions of the essential set of \(w\) are written out, and the rectangle of the plot of \(w\) whose points each one counts is outlined in its color.
The ladder
Each condition implies the one below it. Three of the steps hold for every variety, and two more only for Schubert varieties. At the bottom is every Schubert variety. Each rung gives its pattern criterion, how many Schubert varieties it has at the length chosen, and the permutations of the shortest length on it and not on the rung above: click one to see it up close. In a pattern's picture, box \((i, j)\) is the cell between its \(i\)-th and \((i+1)\)-th points from the left and its \(j\)-th and \((j+1)\)-th points from the bottom, counted from \((0, 0)\) at the bottom left. A shaded box holds no point of \(w\), and a region marked 1 holds at least one.
What each condition says
Each condition but being defined by inclusions is local: it holds at a point or fails there, and the points where it fails form a closed set, for being factorial because finitely many divisors span the divisor classes of \(X_w\). On a Schubert variety that set is kept by \(B\), so it is a union of Schubert varieties, and the condition holds everywhere once it holds at the reference flag, the most singular point.