6 1. OVERVIEW preimages under the Wronski map. These results concern the complex Grassman- nian and complex projective space. Boris Shapiro and Michael Shapiro made a conjecture in 1993/4 about the Wronski map from the real Grassmannian to real projective space. Theorem 1.9. If the polynomial Φ Pmp has only real zeroes, then every point in Wr−1(Φ) is real. Moreover, if Φ has mp simple real zeroes then there are #m,p real points in Wr−1(Φ). This was proven when min(m, p) = 2 by Eremenko and Gabrielov [46], who subsequently found a second, elementary proof [42], which we present in Chapter 11. It was finally settled by Mukhin, Tarasov, and Varchenko [104], who showed that every point in the fiber is real. We sketch their proof in Chapter 12. The second statement, about there being the expected number of real roots, follows from this by an argument of Eremenko and Gabrielov that we reproduce in Chapter 13 (The- orem 13.2). It also follows from a second proof of Mukhin, Tarasov, and Varchenko, in which they directly show transversality [106], which is equivalent to the second statement. This Shapiro Conjecture has appealing geometric interpretations, enjoys links to several areas of mathematics, and has many theoretically satisfying gener- alizations which we will discuss in Chapters 10, 11, 13, and 14. We now mention two of its interpretations. Example 1.10 (The problem of four lines). A geometric interpretation of the Wronski map and the Shapiro Conjecture when m = p = 2 is a variant of the classi- cal problem of the lines in space which meet four given lines. Points in Gr(2, C3[t]) correspond to lines in C3 as follows. The moment curve γ in C3 is the curve with parameterization γ(t) := (t, t2,t3) . A cubic polynomial f is the composition of γ and an affine-linear map C3 C, and so a two-dimensional space of cubic polynomials is a two-dimensional space of affine- linear maps whose common kernel is the corresponding line in C3. (This description is not exact, as some points in Gr(2, C3[t]) correspond to lines at infinity.) Given a polynomial Φ(t) of degree four with distinct real roots, points in the fiber Wr−1(Φ) correspond to the lines in space which meet the four lines tangent to the moment curve γ at its points coming from the roots of Φ. There will be two such lines, and the Shapiro Conjecture asserts that both will be real. It is not hard to see this directly. Any fractional linear change of parameteriza- tion of the moment curve is realized by a projective linear transformation of three- dimensional space which stabilizes the image of the moment curve. Thus we may assume that the polynomial Φ(t) is equal to (t3 t)(t s), which has roots −1, 0, 1, and s, where s (0, 1). Applying an affine transformation to three-dimensional space, the moment curve becomes the curve with parameterization (1.6) γ : t −→ (6t2 1, 7 2 t3 + 3 2 t, 3 2 t 1 2 t3) . Then the lines tangent to γ at the roots −1, 0, 1 of Φ have parameterizations (5 s, −5 + s, −1) , (−1, s, s) , (5 + s , 5 + s , 1) s R . These lie on a hyperboloid of one sheet, which is defined by (1.7) 1 x1 2 + x2 2 x3 2 = 0 .
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