ALGEBRAIC SURFACES, LECTURE 7
LECTURES: ABHINAV KUMAR
1. Ruled Surfaces (contd.)
As before, we have a short exact sequence 0 L E M 0 for any
vector bundle of rank 2 on a curve B . Let deg (E ) = deg (2 E ) = deg Ldeg M ,
hi (E ) = dim H i (B, E ). We can
ALGEBRAIC SURFACES, LECTURE 12
LECTURES: ABHINAV KUMAR
Today we will prove the uniqueness of minimal models of non-ruled surfaces
(in characteristic 0) and talk about the characterization of ruled surfaces.
Theorem 1 (Grothendieck-Cartier). In characteris
ALGEBRAIC SURFACES, LECTURE 3
LECTURES: ABHINAV KUMAR
1. Birational maps continued
Recall that the blowup of X at p is locally given by choosing x, y mp , letting
U be a suciently small Zariski neighborhood of p (on which x and y are regular
functions tha
ALGEBRAIC SURFACES, LECTURE 13
LECTURES: ABHINAV KUMAR
1. Classification of Ruled Surfaces (contd.)
Recall from last time that we had (K D) < 0 for some eective divisor D,
and we wanted to show that X was ruled.
Step 1: There is an ample H s.t. (K H ) <
ALGEBRAIC SURFACES, LECTURE 8
LECTURES: ABHINAV KUMAR
1. Examples
1.1. Linear systems on P2 . Let P be a linear system (of conics, cubics, etc.)
on P2 and : P2 P PN the corresponding rational map. The full linear
=
system of degree k polynomials has dimen
ALGEBRAIC SURFACES, LECTURE 4
LECTURES: ABHINAV KUMAR
We recall the theorem we stated and lemma we proved from last time:
Theorem 1. Let f : X S be a birational morphism of surfaces s.t. f 1 is
g
not dened at a point p S . Then f factors as f : X S S wher
ALGEBRAIC SURFACES, LECTURE 11
Recall from last time that we dened the group scheme PicX over k as well
as the group scheme Pic0 , which is the connected component of 0 (i.e. OX ) in
X
PicX (and is a proper scheme over k ). Now, let L be a line bundle in
ALGEBRAIC SURFACES, LECTURE 2
LECTURES: ABHINAV KUMAR
Remark. In the denition of (L, M ) we wrote M = OX (A B ) where A and B
are irreducible curves. We can think of this as a moving lemma.
1. Linear Equivalence, Algebraic Equivalence, numerical
equivale
ALGEBRAIC SURFACES, LECTURE 6
LECTURES: ABHINAV KUMAR
Corollary 1. Assume that all the closed bers of the morphism : X B
are isomorphic to P1 (i.e. is smooth and the bers have arithmetic genus 0;
or say that : X B is geometrically ruled). Then there exist
ALGEBRAIC SURFACES, LECTURE 10
LECTURES: ABHINAV KUMAR
Recall that we had left to show that there are no surfaces in characteristic
p > 0 satisfying
(1) Pic (X ) is generated by X = OX (K ), and the anticanonical bundle is
ample. In particular, X doesnt h
ALGEBRAIC SURFACES, LECTURE 1
LECTURES: ABHINAV KUMAR
1. Introduction
This course concerns algebraic surfaces, which for our purposes will be projec
tive and non-singular over a eld k . Usually, we will assume k is algebraically
closed. The simplest examp
ALGEBRAIC SURFACES, LECTURE 5
LECTURES: ABHINAV KUMAR
1. Examples
(1) If S Pn , p S , then projection from p gives a rational map S Pn1
dened away from p extending to Blp S = S Pn1 . For instance, if
2
Q is a smooth quadric in P , we get a birational map
Homework 1, 18.727 Spring 2008
1. Do the blowups necessary to reslove the the du Val singularities:
(a) A4 : x2 + y 2 + z 5 = 0,
(b) D5 : x2 + y 2 z + z 4 = 0,
(c) E6 : x2 + y 3 + z 4 = 0,
(d) E7 : x2 + y 3 + y z 3 = 0,
(e) E8 : x2 + y 3 + z 5 = 0.
2. Sho
18.727 Homework 2, Spring 2008
1. Show that every Enriques surface has an elliptic or quasielliptic bration. (Hint: show that it has an
indecomposable curve of canonical type.)
2. If X is a K3 surface with an elliptic bration, show that the base must have