# A peek through the imaginary looking glass

Plymouth, December 2018

# Before starting

• Thank you very much for the invitation

# Recalling projective geometry

The set of lines through a point.

# Recalling projective geometry

Parametrized by the points in a line.

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

# Through the looking glass... to other dimensions

From "Alice through the looking glass"

# Lines

$$y = \lambda x$$

# Lines

$$y = \lambda x$$

Is homeomorphic to the $$x$$ axis: the complex plane.

# 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

# Higher degrees: elliptic curves

$$y^2 = x^3-x$$

# Higher degrees: elliptic curves

$$y^2 = x^3-x$$

# Higher degrees: elliptic curves

$$y^2 = x^3-x$$

# And so on...

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

# 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
• Can be seen in $$\mathbb{R}^3$$ by stereographic projection

# Picard-Lefschetz theory

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

Questions?