u.ewhy of cohomology of complex Complete variables e da dingo e g - - PDF document

SLIDE 1

Computing syzygies

Juliette Bruce

• falgebraicvarieties

Where I

work and live occupies the ancestral lands of

the 9

Lo Chunk

people

and where I

am giving this talk from is

less than 2 blocks

away from where TonyRobinson was murdered by the police

Black andIndigenous Lives Matter

Tonight tomorrow marks

the 51stanniversary of the start of

the Stonewall

Riot Revolution

a pivotal moment in thefight for LGBT

t equality

deadby Black

Brown Indigenous transwomen of color FlappyPride

Black and Indigenous LGBT

t Lives Matter

• tee

mm

Fetuses

ao_so

i

era

X

Ipr

B

sm proj

voiety

S

Exox

in

Ix

hang definingideal X EIP S x

SII x

C

SxC Fo F

Fpc

• Fr

O

mingrodedfree res

Brecxepr

mine

III Eis

grade

Betti

s Betti

s syzygies

The syzygies

Betti

s

are extrinsic e.g depend

• nthe

embeddingXElPr

SLIDE 2

Thm

EinGreenLazarsfeld

Let X E IP

be a smoothcurve

if de

X 0 then i Bp.pt X C Pr

1 0

D

O E p E r g

X

b

gonalityof X f

The Betti

s

capture some ofthe intrinsic

geometry of smooth curves

Centrums

2low is the Intrinsicgeometry of higherdim Varieties

eg dimX 2

capturedby their syzygies

at

simpleton

What are the syzygies

Ipetti

s of P2

Leg nextcasewith

a

simplest

geometry

IP

pr

x y 2 1

xd xdy za

a

L d'upleVeronesemap

image veronesesurf

Bpz P2 d

Bpz IP Ipr

Betti

s

• f

d uple

Veronese surface

Betti

s of R

d

R

Exy z

TEqbringTg

Bp2 P2d

dimeTerps

Sx G of

NBA

resolve Sx symbolically via g b methods

Resolve

Cosm S module via Koszalcomplex

Is

e

no s

u.ewhy

variables e da

Complete

dingo

• f cohomology of complex

e g

Computetheranksof matrices

L sparse

so

SLIDE 3

so

But the matrices we are dealing w

• re FLU

E HAIad FEELEHED

Results BruceErman GoldsteinYang

compute the Betti

s

• f P2 for 1 E de 6

0 Ebcd

as well as the corresponding

multi

graded Betti

s

similar

worn

by GrecoMartino

Schur Betti

s

andcostryctgcooes.Demeye.lemr.cn

Boij

SeidenbergDecompositions

Flown

The complexrespects the 23 multigreding onR

Ddifferentials ore blockdiagonal

Sparsenumerical liner algebra

B

e.g.LU

R decompositions

highperformance highthroughput computing

d

supercomputers

hundreds thousandsof computers

workingatonce

e g LHC HumanGenome

as

Smee

https

syzygydota.com

Schur Veronese for M2 Conjecture

Ein ErmaLazarsfeld 5 There exists an explict set ofmonomial syzygies

Ep 2UP d

such thot

BpHIP d

to

D

Epa

CD d

A small subset of very special syzygies is controlling non vanishing

Conjecture Bruce ErmanGoldsteinYong

With Epa P2d

as above i

d w

P syzygies of

dw

Epq P2 d

a

deg 2

dominantSchur

weights