Math 559 MODERN GEOMETRY REVIEW OF LECTURES – XIX October 23 (Thu), 2014 Instructor: Yasuyuki Kachi Line #: 13191. §19. Grassmannians parametrizing r-spaces in Pn — I. Last time we covered the notion of two linear spaces being mutually in general positions. That was clear-cut. On the other hand, there is some subtlely involved in the notion of three or more linear spaces being mutually in general positions. We defined the notion of three linear spaces being mutually in general positions. How about four or more? Here is what’s most important: Suppose you have k number of linear subspaces inside Pn . Call them Λ1 , · · · , Λk . If the sum of their codimensions is less than n, then you can make a clear-cut, indisputable, definition. Namely, you declare that Λ1 , · · · , Λk inside Pn are mutually in general positions precisely when the sum of their codimensions is equal to the codimension of their intersection Π : codimPn Λ1 + codimPn Λ2 + · · · + codimPn Λk = codimPn Π. The subtlety of the matter is, if the sum of the codimensions of Λi s is strictly greater than n where n is the dimension of Pn , then the same definition does not apply. Even in that case, we succeeded in offering an appropriate definition when k = 2 and k = 3. Namely: (i) Two linear subspaces Λ1 and Λ2 inside Pn whose codimensions add up to a number strictly greater than n are said to be mutually in general positions simply if their intersection is empty. (ii) Three linear subspaces Λ1 , Λ2 and Λ3 inside Pn whose codimensions add up to a number strictly greater than n are said to be mutually in general positions if any two out of Λ1 , Λ2 , Λ3 are mutually in general positions in the sense already defined , and moreover, the intersection of Λ1 , Λ2 and Λ3 is empty. 1 It is not quite self-evident how to generalize this to the case of four or more linear subspaces. As it turns out, the problem suddenly becomes non-linear. For example, in P2 , six points are said to be in general positions if they are not lying in a degree 2 curve conics . This is based on the fact any five points inside P2 are lying in some degree 2 curve conics , where the union of two lines is regarded as a degree 2 curve. Making it more general, namely, in Pn instead of P2 , and linear subspaces of mixed dimensions instead of points, would be extremely challenging. So we are not going to pursue it. I can simply point out that the notion of k linear subspaces of Pn being in general positions has to be preserved after going to the dual projective n-space. Rather, today I want to focus on the duality of Pn itself. For that matter, let’s talk about duality of points and hyperplanes as a starter. Duality in Pn – I. Suppose you say the following: “the two projective n-spaces Pn (X0 :X1 :··:Xn ) and Pn (Y0 :Y1 :··:Yn ) are dual to each other”. Then that is a declaration on your part that you let a hyperplane in Pn (X0 :X1 :··:Xn ) Y0 X 0 + Y1 X 1 + · · · + Yn X n = 0 designate Y0 : Y1 : · · · : Yn a point in P (Y0 :Y1 :··:Yn ) , n and vice versa, and moreover, you let X0 : X1 : · · · : Xn a point in Pn (X0 :X1 :··:Xn ) designate X 0 Y0 + X 1 Y1 + · · · + X n Yn = 0 and vice versa. 2 a hyperplane in P n (Y0 :Y1 :··:Yn ) , Repeat: Saying that the two projective spaces Pn (X0 :X1 :··:Xn ) and Pn (Y0 :Y1 :··:Yn ) are dual to each other is a declaration that you enforce the above designations between points and hyperplanes of the two referenced projective n-spaces. The most important feature of the above duality, needless to say, is the ‘incidence principle’. Namely, suppose (a) σ and Π are a point and a hyperplane in Pn (X0 :X1 :··:Xn ) , respectively, and (b) π and Σ are a point and a hyperplane in Pn (Y0 :Y1 :··:Yn ) , respectively. Suppose that, under the duality, the correspondences σ ←→ Σ and Π ←→ π hold. Then σ ∈ Π ⇐⇒ π ∈ Σ. if and only if • That was easy. Now let’s talk about duality involving lines in Pn . In case n = 3, what was dual to a line? Yes, a line. You have two projective 3-spaces that are dual to each other. Then a line in one projective 3-space uniquely designates a line in its dual projective 3-space, called the dual line to the original line. You remember that. Now, suppose n = 4. Then what do you think? You have two mutually dual projective 4-spaces. Then a line in one projective 4-space uniquely designates what object in its dual projective 4-space? Let’s take a poll. ◦ How many of you think that the answer is ‘a point’ ? ◦ How many of you think that the answer is ‘a line’ ? ◦ How many of you think that the answer is ‘a 2-plane’ ? ◦ How many of you think that the answer is ‘a 3-plane’, or ‘a hyperplane’ ? — I see. The answer is, ‘a 2-plane’. How come? Well, roughly speaking, a 2-plane inside 3 P4 (Y0 :Y1 :Y2 :Y3 :Y4 ) is the intersection of two distinct hyperplanes. Those two distinct hyperplanes designate two distinct points inside P4 (X0 :X1 :X2 :X3 :X4 ) . The line dual to that 2-plane is the ‘unique’ line passing through those two points. Of course, this only makes sense when you see the proof that the line passing through a given pair of distinct points in P4 is unique . Do you know where this is going? Yes? So, in the same token, let n be arbitrary. Then an object inside Pn (Y0 :Y1 :··:Yn ) dual to a line inside Pn (X0 :X1 :··:Xn ) is what? Yes, it is a codimension 2 plane. So, a line inside Pn (X0 :X1 :··:Xn ) ←→ dual a codimension 2-plane inside Pn (Y0 :Y1 :··:Yn ) . You probably find the following plausible: an r-plane inside ←→ dual Pn (X0 :X1 :··:Xn ) a codimension r + 1 -plane inside Pn (Y0 :Y1 :··:Yn ) . But the real question is, of course, how you make this correspondence precise the uniqueness of the line in the case r = 1 , and how you prove that this is indeed a one-to-one correspondence. You probably know what I am going to say next. This is exactly where the notion of Pl¨ ucker coordinates plays a pivotal role. So far we only know Pl¨ ucker coordinates of lines in P3 . How do you go about Pl¨ ucker coordinates of lines in P4 , P5 , etc., or more generally, Pl¨ ucker coordinates of linear r-spaces in Pn ? The truth is, making all these definitions is feasible. There is an established theory of Pl¨ ucker coordinates of linear r-spaces in Pn . It is well-known to experts. For fixed positive integers r and n, with r < n, the set of Pl¨ ucker coordinates n of linear r-spaces inside P becomes a ‘variety’ whose precise meaning being put aside for the moment , which is once again called ‘the Grassmannian variety’. The notation for it is Gr r, Pn . Gr r, Pn parametrizes all linear r-spaces inside Pn . We will give an overview of this topic, starting the next lecture. 4

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