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