A peek through the imaginary looking glass

Miguel Marco

Plymouth, December 2018

Before starting

  • Thank you very much for the invitation
  • Please ask questions

Recalling projective geometry

The set of lines through a point.

Recalling projective geometry

Parametrized by the points in a line.

Recalling projective geometry

Except for one.

Recalling projective geometry

Solution: add one point to the line

  • The projective line is the line completed with one point at infinity.

From Euclid:

A line cuts a circle in two points:

From Euclid:

A line cuts a circle in two points:

From Euclid:

A line cuts a circle in two points... or not:

Algebraicaly:

\[(x-2)^2+y^2-2 = 0\]

\[y=\lambda\cdot x\]

Algebraicaly:

\[(x-2)^2 + (\lambda \cdot x)^2 -2 = 0\]

Algebraicaly:

\[(\lambda^2+1)x^2 - 4x + 2 = 0\]

Algebraicaly:

\[x = \frac{2\pm\sqrt{2-2\lambda^2}}{\lambda^2+1}\]

  • \(\lambda \leq 1\): real solutions
  • \(\lambda \gt 1\): complex solutions

Algebraicaly:

Algebraicaly:

Through the looking glass... to other dimensions

From "Alice through the looking glass"
© copyright The Walt Disney Company.

So, how do objects look like in this extra dimensions?

Lines

\(y = \lambda x\)

Lines

\(y = \lambda x\)

Is homeomorphic to the \(x\) axis: the complex plane.

Lines

Projective Lines

Conics

Conics

Conics

Conics

Conics

Conics

What happens around the discriminant?

\(y = \sqrt{x}\)

What happens around the discriminant?

  • \(x = e^{2\pi i\theta}\)
  • \(y = \pm e^{\pi i\theta}\)
  • As \(x\) makes the full turn, \(y\) makes half a turn

What happens around the discriminant?

What happens around the discriminant?

Stairs between the discriminant

Stairs between the discriminant

Stairs between the discriminant

Stairs between the discriminant

Projective conics

Projective conics

Projective conics

Higher degrees: elliptic curves

\(y^2 = x^3-x\)

Higher degrees: elliptic curves

\(y^2 = x^3-x\)

Higher degrees: elliptic curves

\(y^2 = x^3-x\)

Projective cubics

And so on...

Theorem: a smooth complex projective curve of degree \(d\) has the topology of a surface of genus \(\frac{(d-1)(d-2)}{2}\)

But... what about singular curves?

Let's look closely

\(x^3-y^2=0\)

Let's look closely

\(x^3-y^2=0\)

Let's look closely

\(x^3-y^2=0\)

Local topology of singularities

  • It is the topological cone over the intersection with the boundary of the Milnor ball
  • So... what is this intersection?

Local topology of singularities

  • The boundary of the Milnor ball is \(\mathbb{S}^3\)
  • The part of the complex curve inside the disk is a piece of surface
  • The intersection with the boundary is the boundary of a piece of surface:
    • a curve without boundary.

Closed curves in \(\mathbb{S}^3\)

  • With one component: knots
  • In general: links
  • Can be seen in \(\mathbb{R}^3\) by stereographic projection

Milnor's trick: see them as a limit of smooth cases

Milnor's trick: see them as a limit of smooth cases

Pushing the Milnor fibre to \(\mathbb{S}^3\)

Pushing the Milnor fibre to \(\mathbb{S}^3\)

Milnor fibre

Milnor fibration

Picard-Lefschetz theory

Theorem: The singular curve is the result of collapsing some circles (vanishing cycles) on the Milnor fibre.

Thank you

Questions?

Zariski's point of view: braids