Knuth 22 O C - Lecture 16

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May 29 ,2020

Riemann - Roch & Weierstrap problems- - - - -

- -- - -

Divisors /Principal divisors

Sheaves attached to divisors

Rephrasing of Riemann- Roch (Weserstraps via sheaves .

II Divisors D E X divisor if

D= pc€ hp Ep] where np c- 21.

& { p : np fo ) is locally finiteX compact ⇒ finite

.

Eixample X = CT, D = 2 . [o] + 3 . [a] - 5 [i]

.

→ deg D - o

E = [o] - 2 - [ is] → deg E =-n .

b + E = 3 [o] t E-T - o- [ i]

.

Bernard II D. 20 effective if mpzo .

If sums : D= Emp EP]⇒ Die = Ecnptmp) Ep]

.

E = [ Mp Cp]

icy restrictions : Dfw = I rip [p]

peu.

II sheaf Dios → x

u - DI ( u) = { divisors over U } ..

II degree ,X compact .

deg D = E "p

Principal divisors f meromorphic . on X , f # o .-- - -

gorder

←order

• dir f = I mz ( z] - Emp Cp]2- Zero

p pole

• claw C f-g.) = div f t drug

Eixample I X = E.

f- = 2-( s - 2)

3

di- f = [o] - 3 [i] it 2 [A].

.2

At -,z -_ Yw

.

⇒ f- = %- = = ⇒ •-do f = 2 .

( r - yw)3 Cw- if

Nok deg dwf =o .

x = E : f = ' 2-

..

( 2 - b,) . . .

CZ - bm)

check : de- f = €,

Cai] - §.

[ b,]

,t (m - n>

.

C-T.

⇒ oleg du f = o .

IT, X = % : any elliptic function

We saw # Zero =# poles counted w/multiplicity =3 of - gdw f- = o.

I

Enact Curll not use) for all x compact ,

d- gdu f = o

Two problems involving divisors- - -

- -

A.

Weier strap, problem Isany divisor on X principal ?

--

⇐S F f meromorphic north prescribed Zeros d.poses .

?

Answer /"" compact x

- -

"

\ compact . X

Remark- -

II non - compact X E Q .

l Math 220 B).

compact .For X = E we saw we need degD= • .

Even if deg D =o , the answer may be No if X fE

.

Rephrase Weierstrass--

-

Let F-→ x .

Write Fcx) = rcx,F) = Hocx

,I)

.

txample I = 0 ,x compact => Ho Cx , G) = a .

two streams I = At#= meromorphic functions ¥0 on any component

} = Dig = sheaf of divisors

Weier strap asks- - -

Ho Cx,M* ) → Ho ( x

,DIE)

. surjective ?

f - divf

B. Riemann - Roch problem- - - - -

Fx £. . - -

2-n , Pe - - - pm .

E X, fun . - - In ,

ve - - - Um 20 integers

Question - Describe thespace of meromorphic functions on X. with

- -

'

• Zeros of given order 2mi at Zi .{. poles with gwen order I Vi

at pi & noother poles

Rephrase-- -

D= - E Mitzi] + Ev.-Ep ,] :

Describe

{ f meromorphic ,

div f t D Zo}.

Eexazpte X = % . D= d- Co]

dem Xd =D

Vd = L l, js , 2b

'

,. . . js

'd-"z ( last time)

.

Sheaves associated to divisors- - - - -

D divisor.

The sheaf'

①×(D) is defined by

U - { f ¥0 meromorphic , du f t Dtu Zo} u fo}.

if u connected.

Question Describe the global sections Ho (x, Ox CDT)

.--

Outline of the answer I→ x sheaf , X Riemann surface- - -- - -

⇒ Sheaf cohomology

IAI define HP Cx,t) for pz o .

We have

Ho Cx,F) = rcx

,Is = FCX)

.

'¥ X (x ,F) = [ C-DP dim HPCX

,F) ! we" - defined ?

II. Answer to Weierstrap :

H-

( x , M*) → Ho (x,Div s -yeah-

*We will see that if H

'(x , O 3=0 ⇒ yes

Answer to Riemann- Roch

. X compact .

• y = XCX ,G) = dam Ho Cx

,G) - dim H' Cx

,G) = r - g-

g- genus

• X ( x , Ox (DD = X t deg D-

=>i dim H°(x

, Ox Cbs) z y e- deg D .

The difference is H' ( x , Ox CD)) .

Gustav Roch (1839-1866)

Crelle's Journal

On the number of arbitrary constants in algebraic functions

Sheaves on a Riemann surface- - - - - -

( summary)

• G = holomorphic• ①

*

= nowhere zero holomorphic functions

• M = meromorphic functions

*• M = meromorphic functions not identically o

.

• To ?= smooth functions

• I = locally constant functions

• D2've = sheaf of divisors-

• G× Cb)

-

④ These sheaves solve different problems via Shea f cohomology

④ they interact with each other

She- f cohomology- - -

Gg at II define cohomology groups .

I learn how to compute them .

Fundamental Example- -- . .

• complexes . . .

→ Ak d- Ak" d- Ak-12→ . . - A.

d2= O

• cohomology

71k (A. ) =

Ker d : Ak- akin- - -

Im d : Ak-'→ Ak

• short exact sequence o - Ai - B.

- c'

→ o

⇐ o → Ak → Bk- Ck→ o,

gives →Hk ( ai ) - FIKCB. ) → ftkcc. ),-

& Hk" CA.

) → . . .

For sheaves- - -

• → F - S - Th → o- C to be defined)

.

goes → it" (x

,F) → H'

'

Cx, 9) → HP Cx, Te) .

-S H't

' (x. Is → H'"'

(x,as) - H'"

'

(x,Je) >

#

Functional Operations with Sheaves

-- - - -

-

II morphisms

Il exact sequences .

ktrfhimes -

L : I → G morphism of sheaves consists sis

tu '

- Icu) → Sfu). compatible with restrictions

Lu.

Fcu) - Scu) U IV.

fee 2 tentLv

Fhs - Slv)

Renard If a : I- S then xp : Fp - Sp for all pex .

-

Fp → a Cf)p .

well - defined.

x p .

Exact sequences o → I → S → Je → o is exact if- - -

o → Ip → Gp → yep - o.

is exact . t PEX .

Exponential segue? locally constant integer - valued functions.

- ---

o - Z.- O - 0*-1

.

2T if .f - e

Claim This sequence is exact .

• → I → G → G 't → I.exact

.- p p .

-

÷ Ggg → g exists in - empty connected

open sets around u so

surgeeternity follows on stalks .

FAI : o → Z Cx) → 01×3 → G*Cx) → I exact.

⇒ Surgeeternity doesn't hold on open sets only on stalks.

Why ? X = a.

' to}.

• → z - G → G* → s.

1- Logz - Z251

'

-

cannot be deferred on a s { o ).