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Introduction to Schubert Varieties Lms Summer School, Glasgow 2018 INTRODUCTION TO SCHUBERT VARIETIES LMS SUMMER SCHOOL, GLASGOW 2018 MARTINA LANINI 1. The birth: Schubert's enumerative geometry Schubert varieties owe their name to Hermann Schubert, who in 1879 pub- lished his book on enumerative geometry [Schu]. At that time, several people, such as Grassmann, Giambelli, Pieri, Severi, and of course Schubert, were inter- ested in this sort of questions. For instance, in [Schu, Example 1 of x4] Schubert asks 3 1 How many lines in PC do intersect 4 given lines? Schubert's answer is two, provided that the lines are in general position: How did he get it? And what does general position mean? Assume that we can divide the four lines `1; `2; `3; `4 into two pairs, say f`1; `2g and f`3; `4g such that the lines in each pair intersect in exactly one point and do not touch the other two. Then each pair generates a plane, and we assume that these two planes, say π1 and π2, are not parallel (i.e. the lines are in general position). Then, the planes intersect in a line. If this happens, there are exactly two lines intersecting `1; `2; `3; `4: the one lying in the intersection of the two planes and the one passing through the two intersection points `1 \`2 and `3 \ `4. Schubert was using the principle of special position, or conservation of num- ber. This principle says that the number of solutions to an intersection problem is the same for any configuration which admits a finite number of solutions (if this is the case, the objects forming the configuration are said to be in general position). Unluckily, the notion of general position is ambigous, and caused several mistakes in solution counting. Making enumerative geometry rigourous was the 15th of Hilbert's problems and at the origin of the Schubert calculus. We will not be able to discuss this into details 1.1. Schubert's question translated into modern language. Recall that the complex projective space n is defined as the quotient of n nf(0; 0;:::; 0)tg PC C by the equivalence relation ∼ given by 0 1 0 1 0 1 x1 x1 x1 B . C B . C B . C n t @ . A ∼ λ @ . A for any λ 2 C n f0g; @ . A 2 C n f(0; 0;:::; 0) g: xn xn xn 1 2 MARTINA LANINI t n t We denote the equivalence class of a point (x1; : : : ; xn) 2 C n f(0; 0;:::; 0) g by 2 3 x1 6 . 7 4 . 5 : xn The reason why we are using column vectors instead of row vectors will become clear later in Exercise 3.5 Notice that, by definition, a point in n is a line through the origin in n PC C (hence, it is nothing but a 1-dimensional subspace). Let's get back to Schubert's question. He was considering lines in 3 , which PC 4 are nothing but 2-dimensional subspaces of C (see x1.1.1). n Definition 1.1. The set of k-dimensional subspaces of C is denoted by Gr(k; n). We will refer to it as the Grassmannian or Grassmann variety. As the name suggests, Gr(k; n) is more than a set: it is a projective algebraic variety. We will get back to this in the next section. Exercise 1.1. Show that for any m there exist k; n such that Gr(k; n) = m. PC The four given lines are hence four points W1;W2;W3;W4 2 Gr(2; 4) and Schubert's example amounts in determining #fV 2 Gr(2; 4) j V \ Wi 6= 0 for i = 1; 2; 3; 4g: For i = 1; 2; 3; 4, define (1) Yi := fV 2 Gr(2; 4) j V \ Wi 6= 0g; T4 so that we are looking for # i=1 Yi. Each one of the Yi's is a Schubert variety: a variety defined by incidence relations (see Example 2.6). 1.1.1. Little digression on projective geometry. While discussing Schubert's enu- merative geometric question, we looked at lines in 3 , and we then said that PC 4 they are nothing but (images) of 2-dimensional subspaces in C . In general, a n n+1 k-space in P is the image of a k + 1-dimensional subspace of C . A hyperplane (or an n − 1-space) in n is the zero set of a homogeneous PC polynomial of degree 1: a0x0 + a1x1 + : : : anxn = 0; n where (a1; : : : ; an) 2 C n f(0;:::; 0)g. A k-space in n is the intersection of n − k hyperplanes: PC 8 a x + a x + ::: + a x = 0 > 1;0 0 1;1 1 1;n n <> a2;0x0 + a2;1x1 + ::: + a2;nxn = 0 (2) . > . > : an−k;0x0 + an−k;1x1 + ::: + an−k;nxn = 0 such that the rank of the matrix (ai;j) is maximal (i.e. rk(ai;j) = n − k). At this point the correspondence n n+1 fk-spaces in PCg $ f(k + 1) − dimensional subspaces in C g 3 should be clear: we know that a homogeneous system of equations in m- variables over C, with associate coefficient matrix A, admits a solution space of dimension m − rk(A), so that if we now look for the solution space of the n+1 system (2) in C , we see that it has dimension n + 1 − (n − k) = k + 1. 2. The case of the Grassmann variety 2.1. Basics on varieties. Definition 2.1. An affine (algebraic) variety is the set of solutions to a set of n polynomials in F for some field F. Example 2.1. For example, the graph of y = x2 − x is an affine variety in 2 R , since it is the set of all points (x; y) annihilating the polynomial f(x; y) = y − x2 + x. Definition 2.2. A projective (algebraic) variety is the common zero locus in n of a finite set of homogeneous polynomials in n. PF PF Observe that if f(x0; : : : ; xn) is homogeneous, then deg(f) f(λx0; : : : ; λxn) = λ f(x0; : : : ; xn); for any λ 2 F n f0g; so that f(x0; : : : ; xn) = 0 if and only if f(λx0; : : : ; λxn) = 0 for any λ 2 F n f0g. Hence it makes sense to talk about zeroes of a homogeneous polynomial in n. PF We will focus on the case F = C, the field of complex numbers, and explain why Gr(k; n) is a complex projective variety. 2.2. A canonical representative in Matn×k. We denote by Matn×k the set of n × k matrices with coefficients in C. n Let (e1; : : : ; en) be the standard basis of C (all entries of ei are zero, but the i-th which is a 1). Given a subspace U 2 Gr(k; n), we want to associate with it a matrix MU 2 Matn×k in a canonical way. First of all, we can choose a basis for U, say (b1; : : : bk), and can write all bi's in terms of the standard basis. In this way we obtain k column vectors, that we arrange into an n × k matrix (having as columns the bi's ). Notice that we can perform column operations to get another matrix, corresponding to another basis of the same subspace, and hence the n same point in Gr(k; n). It follows that any k-dimensional subspace U of C is 2 represented by a matrix MU in column echelon form . If, moreover, we assume that the pivots are ordered left to right, top to bottom, then such a matrix is unique and we denote it by MU . Example 2.2. Let us assume k = 2, n = 4 and denote by (e1; e2; e3; e4) the 4 standard basis of C . If we assume (3) U = spanCf8e1 + 4e2 + 2e3; 5e1 + 2e2 + e3g; 2A matrix is in column echelon form if any column has a pivotal 1, i.e., a 1 such that it is the first non-zero entry looking from the bottom of such a column and the first non-zero entry looking from the right of the column to whom it belongs. 4 MARTINA LANINI then, 0 8 5 1 0 −2 5 1 0 1 5 1 0 10 1 B 4 2 C B 0 2 C B 0 2 C B 0 2 C B C ! B C ! B C ! B C = MU @ 2 1 A @ 0 1 A @ 0 1 A @ 01 A 0 0 0 0 0 0 0 0 2.3. Schubert cells. Definition 2.3. Let U 2 Gr(k; n), and let MU 2 Matn×k be the representing matrix. The index set of the rows where the pivotal ones are is called the position of U, and denoted by pos(U). Notice that by definition of MU , the position is a strictly increasing sequence of k numbers choosen from the set f1; : : : ; ng. Example 2.3. In Example 2.2, we have pos(U) = f1; 3g. k We notice right away that the number of possible positions is n , that is the cardinality of set of the subsets of f1; : : : ; ng with k elements. Definition 2.4. Let i = (i1 < i2 < : : : < ik) with ij 2 f1; : : : ; ng, then the Schubert cell Ci is Ci = fU 2 Gr(k; n) j pos(U) = ig = fU 2 Gr(k; n) j dim(U \ spanfe1; : : : ; erg) = j; 8ij ≤ r ≤ ij+1 − 1g; where i0 := 0 and ik+1 = n by convention.
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