1

Catalogue

1. 1. Seminar
   1. 1.1. Plan
      1. 1.1.1. Part I
      2. 1.1.2. Part II
   2. 1.2. First
      1. 1.2.1. starter
      2. 1.2.2. section 1

Seminar

Plan

Part I

• parametrizing morphisms (“Bend-and-break” producing rational curves)
• Unirules and Ratinally connected varieties (“smoothing tree of rational curves”)
  Reference

1. higher dimensional algebraic varieties, Debarre
2. Rational curves on algebraic varieties, Kollar

Part II

• Vanishing theorem
• Cone theorem
• Base point free theorem
• Singularities
  Reference

1. Brational geometry of algebraic varieties, Kollar-Mori
2. 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