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 very 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?

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 \(H = V(x_i)\).
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