Tags: #study-guides

Divisors

Definitions

• Types of divisors:

  • What is a prime divisor?
  • What is a principal divisor?
  • What is a Weil divisor?
  • What is a Cartier divisor?
  • What is an effective divisor?
  • What is a reduced divisor?
  • What is an ample divisor?
  • What is a Unsorted/ample divisor?
  • What is a nef divisor?
  • What is a big divisor?
  • What is a SNC divisor?
  • What is a simple normal crossings (SNC) divisor?

• What is the divisor of zeros and poles of \(f \in k(X)^{\times}\)?

• What is the pullback of a divisor?

• What is an invertible sheaf?

• What is the invertible sheaf associated to a divisor?

• What does it mean for divisors to be linearly equivalent?

• What does it mean for two divisors to be numerically equivalent?

• What is the divisor class group?

• What is the Picard group?

• What is the divisor associated to a section of a bundle?

• For \(D\) a prime divisor and \(p\in \mathop{\mathrm{supp}}D\), what is the multiplicity of \(p\) in \(D\)?

• For \(D\) a divisor, what is \({\mathcal{O}}_X(D)\)?

• What is a polarization?

• What is a variety of general type?

• What is the ramification divisor of a surjective morphism \(f:X\to Y\) of smooth projective curves?

  • What is the ramification index?
  • What is a branch point?
  • What is the branch locus?

• What is a linear system?

  • What is a complete linear system?
    • What is the complete linear system associated to a section?
  • What is the base locus of a linear system?
  • What does it mean for a linear system to be base point free?

• What is the Riemann-Roch space?

• What is a hyperplane section?

Results

• When do Weil and Cartier divisors coincide?

Problems

• Show that the hyperplane sections of a projective variety \(X\) form a base point free linear system of effective divisors on \(X\).
  • Show that if \(X\) is normal, then a generic element of this system is a smooth and reduced divisor.
• Show that \({ \operatorname{Cl}} ({\mathbb{A}}^n_{/ {k}} ) = 0\)
• Show that \({ \operatorname{Cl}} ({\mathbb{P}}^n_{/ {k}} ) \cong {\mathbb{Z}}\), generated by the class of \(H = V(x_i)\).