Knuth 22 O C - Lecture 16
- - - -
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 ).