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Seminar

Plan

Part I

  • parametrizing morphisms ("Bend-and-break" producing rational curves)

  • Unirules and Ratinally connected varieties ("smoothing tree of rational curves")
    Reference

  • higher dimensional algebraic varieties, Debarre

  • Rational curves on algebraic varieties, Kollar

Part II

  • Vanishing theorem
  • Cone theorem
  • Base point free theorem

  • Singularities
    Reference

  • Brational geometry of algebraic varieties, Kollar-Mori

  • Algebraic geometry and finitely generation, Kawamata (translated by Chen Jiang)

First

starter

Mori: Characterization of projective space.

Hartshorne's conjecture:

Let \(X\) be a smooth projective variety of dimension \(n\) over algebraically closed field of and Characteristic. Then \(T_X\) ample iff \(X\cong \mathbb{P}^n\).

Note that a vector bundle \(E\) is ample (nef) if \(\mathcal{O}_{\mathbb{P}(E)}(1)\) is ample (nef).

section 1

Def:
\(f: \mathbb{P}^1\to X\) a nonconstant morphism is called a rational curve on \(X\).

Grothendieck's moduli: \(Mor(Y,X)\)

e.g. consider \(Mor(\mathbb{P}^1,\mathbb{P}^n)\), a map \(\mathbb{P}^1\to \mathbb{P}^n\) is given by

$$ [u:v] \mapsto [F_0(u,v):\cdots:F_n(u,v)] $$
where \(F_i\) are homogeneous polynomials