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 .