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:
Through the looking glass... to other dimensions
Lines
\(y = \lambda x\)
Lines
\(y = \lambda x\)
Is homeomorphic to the \(x\) axis: the complex plane.
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
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 \(\frac{(d-1)(d-2)}{2}\)
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
Picard-Lefschetz theory
Theorem: The singular curve is the result of collapsing some circles (vanishing cycles) on the Milnor fibre.
Zariski's point of view: braids