Enumerative Geometry 06 Gromov Witten

Intro

• How many points in intersection of two lines?
• How many points in intersection of two conics?
• Given \(3d-1\) points in general position, how many rational curves of degree \(d\) pass through?

Intersection Theory

Background

• Affine Space
• Projective Space
• Homogeneous coordinates
• Algebraic variety in projective space is vanishing of homogeneous polynomial.
• \(P^2\) . curve (vanishing polynomial) is a line if degree 1, conic if degree 2, cubic if degree 3, …, and so on.

QnA

• Given a polynomial of degree \(n\), how many points on the number line (could be complex, depending on the algebraically closed field)?
  • If some caution, then \(n\).
• How many points in intersection of two lines?
  • 1 if considered in \(P^2\).
  • The point at infinity is the intersection of two parallel lines.
• How many points in intersection of two conics?
  • 0 or 1 or 2 or 3 or 4. Change question slightly: answer always 4.
  • Bezout
• Alg closed
• Cycle : Linear combination of subvariety
• Rational equivalence Chow ring
• Transversal
• co-dimension
• degree
• Bezout

Tomorrow

• 3264 and all that
• Moduli space