stabilityofnormal.bedesofJwHXBriHNoetherspace ur

based on joint work with

Izzet Coston and Eric Larson

k k char Chl arbitraryin

O The goal of this talk is to

say something about the geometryof gyral curves

call thesef CEE HI Brill Nogthere

rH d ray degd

I iiis mi i.si geb S1 d the al b dl

by studying their normal bundleso To f Fpr Nc o

which controls deformation theory

HoCC Ncfirst order defsof f Ces Rr

H1Cc Nc obstructions to liftingfirstorder deformations to all

orders

By Hardr Narasinhamthory it's naturalto ask if Nc is

Stable semi stable

Def A vector bundle E on a smooth

curve C is called semi stable ifH sub bundles F c E

deff ie F e µ E DIErkF s rkE

If E is not semistable then its unstable2

Ex C R E e Cai

n anglais

E is semistable all aiane equal

stable r I

Stable bundles are the buildingblocks of all bundles

ListabilityresotsExamples of curves w stable normal bundle

Wheur 3andchark

Sacchiero 83 D 6 g 2 stable

EHia 83 stable

Ballico Ellia 84 d 7 g 5 stableD 7 g 4 semistable

FI Ifz 84 asymptotic results if d.gs o

Inanother direction

j Aprodu Farkas Ortega 16 The normal bundleofa general canonical curve is stable once ga't

when

We completely characterize when the normalbundle of a Brill Noether space curve

is stable

theorem Coskun Larson V 120 Let C be a

general BuriNomether curve in of degd and

genus g p Cg r d Z O

fcanonical curves

If ld.glt5kJxhen Ncisunstable

0 d even AND cha_k2Happen good geom reason

If 8 then Nc is

or strictlyF O EE't'Ier doted Semmitable

chark2

Best you could hopefor whengs1Otherwise Nc is t.sbleJ.to

1J

Using fund exact seq

ie Nc 2dtgrk Nc 2

Nc is stable all line subbundleshave deg s zdtg 2

Ho Nc Lv _o

tf line bundles of

SSJ.it stabecasesdeg32dtg I

IcharacteristiczI Assume that is

a rational curve of degree d

ie Nc ad 1

Nc is semistable

se b L s

Ne t Clap lad 1 Claplad4

Inany characteristic we have the Eulersequence on LP

o Of 11 off Test 1 0

Xo X y

ii

exactness here is Eulerrelation

IxiZ 0

On C

o P Ochi OF Nd 1 o

kernel is vile 2 bundle whosefiber is come over Tp C

I p C

At

6Kp Tpc

Incharacteristic2P Oct 1 a Oct 1

where x C is Frobenius

7 Nc is pulled back under a

degree 2 map every term is

even

Every summand of Nc has degreecongruent to d mod 2

If d is odd no obst to

Nc 0 fed c to CfpGd 1

But if d is even most balanced is

Ep Zd QI 2d 2

IArbitrarycharacteristic If d g LEIquad resp 16

thenHo ceased H Octa

din 3121 10 RR dim 9

has a nontrivial kernelC always lies on a quadric

surfacN

undleexuctsequence.co

Nc a Nc Nate o

U 12 all me 10

18 me 15 n 12

V

Nola is dest sub bundle

3 Proofofstabilityidegeneration

Key suppose you have a family II B'tEl

14 I E 3 isY Exc

D Tu Flt ExcE has negand a vector bundle stope

W a destabilizing subbundleFo e to

9restriction to

After an appropriate blowupgeneric fiber

can extend toF E p't E

In this sense stability is an opedconedition in families of nodalcurves

X YFirstidia Restrictions to components of

the central fiber are close to stable

NcLemma 7 E vector bundle on E s t

testableE genefiberEly is semistable is stable

Prod Lef Fo e E o be a subbundle

After possibly passing toa blowup p C E

can extend to

F e p't E

then might havenegslope on Exc

ulFlgene ul Flspeciae Euffly t

nifty

ul Elp tutelyµ E special

ulElgenie

In order to use Lemma 1 need tounderstand

N uylTurn

intrinsically on X

X Nine IRelatedX

µ

t tfii

Hartshorne

such deformations Hirschowitatflowed to have a

newer smooth the pole p in directionnode p of TpY

Notation N uyl N lp Lp N Ias

NxlpjCp sq3

fFirstideai_vokn.need a degeneration

Xu Ys t

Nxuyly is semistable

Lemma2_ In the following two cases

Nye0111100

Gf or 0Hise

f

Y Y a 4secant conic

Nxuyly I 06 OG and LdegX genX

JNxuui.isabetl4Horl3 o

Upshot

lstablIxT_teFIsfaicaeiFiYfto9Tuey.v.twg Oisstabie

deg d I V b on curve of

d 2 degd genusgand genus

iiit

T m

allcasesgswgt.gg

But Nxuylyissemis.to beisa

strontian

Secon dIad Even if Nx y is unstable

use gluing conditionsnodes

Indutify destabilizing sub bundle ofNxvylyShow it cannot agreesufficiently

with a sub bundle of N uylto produce a destabilizingsub bundle of N uy

Second idea work 1 secant linedegeneration

quoteXf v

for each gE8qpX

I

need all a set Ad Zgt3 qNeed a way toinductively decrease Nµy1u a OCI OHd

if

u

Ny Nya Nyan

Nxuyly Ny afp to Nyt p

LemmaI If N uyl pLIis stable then for any curve C specializingto 12 Nois stable

Hardstepofproot

Suppose L is a destabilizing line

qubbundleof Nc that extends over

he central fiber 4isomIo T

RNxuu.ly UNIT IN v ly

HH a HHTX y

p p

jE j y fn.int

Hardestcase ioh passes

thru y

Ted passes thru Mp

LI is a subbundle of

N p

uI hI u

ui t

µ N I 2 MlN 1 1 36

U N uyl tµlNxuy1y Ml Ne

19

These ideas f E reduces Thin 1 to

dig E 612 7 2 6,33 1713 7,4 18,4

52 4 8jg196119714971

tonly nec if wantto seal w chark 3