REV 2019Day I 6 3 2019

C.Berkeschvirl.uasyzygieskQa9EeEaiiEhs

Toms actionI e Cht I

xhtt

Ct F Exo tx tx n t I

I fg eEnt Ft EI st yet IlinethroughE containg I without o

Projectivespace

PqExt IE so

an soywith EI Cgt

I

is

s't t.gg

It helpstocompactity an

lN i fo l 2xD

IIxidi lat

FCI I caId E GGox If Sx

finite sun E f

te Exacts on polynomialstext via

fCtx 2k Exo tx txnYa

Def f is homogeneous ifF kli the same

Thenfly o FgEATwhen o

when f is homogeneous

For XcpnTIX ESffetoteEXEXAMPLETp3X Ii for DGOD

a 0,0

has oridealgeneratedbyor

ICXHxi.xatntxoixdkxoixDcxox.iroXaixixD

P3Y fa find G O D

hasICXHxaixdnlxo xiixdntxo2xa.my

L4XzXoX 2 27 4 47

DEI IES is an ideal if1 It 0 o EI

2 a b c I a c be I3 a EI f ES fa EI

Claims is an ideal

Theidealgeneratedby fi fees isA Sfi fr fIzhifi hiESHilb aTheom Ihs

every ideal is finitelygenerated

DEF An ideal ICS is homogeneous

if ithas ageneratingsetcontainingonlyhomogeneous polynomials

CLAIM ForXcpn I is homogeneous

DEF Given a homogideal F tf Az frins letI EEP f e off EI

fEEP f Lef flat ois a projectivealgebraicarily

EXAMPLEpbxfciooffoioffo.o.MY

V Iwhere xoxoXoX2XXD

AtgomelyGeometric

vegare Algebraic

propertiesof propertiesd SII

esirreducible arm domain

k Ia primeideal

Re sel

a Prove x is an ideal

b Given ICS a homog ideal

set fI FES Fm c BostFEI

thetrade of Icompute figsShow TI is ahomog ideal

c Given X EY EPT showI LX 2 ICY

Given I E J E S homogidealsshow U I Z VCJ

d Find I sit IF I SUEDGren I JES homog ideals

showVCITD VCI outs

NoEe butrelated

FACTSIN CID EX projectre variety

MIND X

DEF Agradedminimalfreeresolutionof SLI I a homog ideal is

Fo se F EE e

such that

it zsfjjPig

where Stj means S with 1 viewedas g in homogeneousdegreejii 9 fit 0 Ex is a complexand in fact more stronglyIli in 9 it a

Ciii coker4 Flinch SII

E Go XD E ti

G all 9 have degree 0bearing in mind thedegree

shift B

EXAMPLEFo

slightestSlDeo

Hilbutissyzygythm X EphSIT X has a minimalgradedfreeresin

to F e Fn 0

oflength n din P

Computing minimalfree resins is no

so easy in general but thereare algorithms involving

Grobnerbases which allow

computers to do it

Vi

Fix T E in EI

Ext x CIN ENCE xt E x Exa In

EXAMPLE r 2 N 5

ax x E5CE E 1 Ctx tik.be aXytaxf

so 4 f I2E E

f Glx Xn

Y3 irrelevantideal determinedbythefan of X

h aboveexample knotS QCXi Xs

definedyet

Xuxa nasik XD

Then we definexfiCNWCBDfx5lnaboveexTpgX

C5iVkxsxa7NXs.Xyya

p p2T.suhedLvpwjeciLonot

Macaulay 2 software demoshowing a resolution of a curve in

IP xp thattook 4 steps

DEF IES an ideal

annCSII Efes f5 of5EDI

DE For YEXa virtual resolution of is

J J2FoGF Fz with

E Pig

VGnnfker.gg o

Vizn.VfimHY

Virtual Hilbert Syzygy TheoremCB.frLh2oI7TXfpMxpn2x xp

X smoothtonic ofdimension 2

Then HY E XS IEDhas a virtual resolution

otlength EdimX

Macaulay 2 showed a virtualresolution of some curve now of

length 2

Methods to make

virtual res

i virtual Of Pair see CBESD

res In Boitake subsets ofgens of I

Macaulay 2 demo

REUPROBLEM 1

a smoothprojectivetonic variety

2 CX a finite setofpointsofX

ten.fm

mxMacaulay2hasdatabasesofsuchX

e g smoothfanotonicVariety rpm

Kleinschmidt m Ldin t

REUExerc ise.dea What is I Kao ai kobD

IPIplb Find a vres of length 2 forstifywheref co D co D

Gio G DG 27 G o CP xp

Use IED amethodWhich a's work for length 2

c LetXbeasmoothfanotoridarietyBDUsethecommandtoricPointswith axXIi gb

intersect 3 90,0the idealsof 0 90,0thesethree 2 11,1

Use C InBoimelhodtomakerresof f 3 points

above