symbolic powers and the (stable ) containment problemNeuchatel -Fribourg - Boen Seminar 15/05/2020

R regular (great working example : R = k En,y, z ) )

I = TI = I,h . . .A In

h= big height of I = m.gg {ht Ii }

The n-th symbolic power Of I -

.

In ' = A fin Rp n R )i -i

= Z EER : SEE IT some SEf VI; }

theorem (Zariski-Nagata ) R = k Ena, . . . , sad] , k perfect fieldI" )=3 f ER : f vanishes to order n along }

the variety defined by I= { f ER :

gait- - - + adf E I forall aet - - - tad e n- I}

ICM ) e Icn )Facts "2) Ine I'm3) If I=C regular sequence) , III

" '

for all he

3 4 5

Exameple I = beer( tiny,⇒ → k# t't]) prime= ( n'- yz , y

'- Nz,za- ray )

of T Tdeg 9 deg 8 deg to

Claim : IK)⇒ 22

2

I a f'- gh =z of ⇒ of e I

'"

-EE

deg 18 = deg 3 tags → ofEff ← deg > 16

I'"e I I'

Note the ideals I defining Ctatb, t' )

have very interesting symbolic provers . EgIn't e RETI is not always fg

(sometimes it is, and generated in degree E 1,2, 3,4 . . . .)

BigQuestions

1) Give generators for I'm2) When is I'm = In ?

3) what degrees does Icn ) live in ?

Containment Indole when is Ica ' e Ib ?

theorem (Ein -Lazarsfeld- Smith, Hochstein-Huneke, Ra - Schwed)2001 2002 2078

Ichn ) E In for all next

Example our prime I defining Ct?th,t) has h=2To E"' e I? But actually I EI?

Question I pane of height 2 in a RLR .Is I

'E Ia ?

Confiture (Harbourne ) Ian- htt 's In for n >e

Fast In char p , Ich't-htt )

E Itt's It for gape .

Counterexample Dummcki - Sternberg- Tutty-aahn'ska,293( Harbourne - Sealeanu,205 )

I = ( Ny"- E )

, yCE- rn ), 2- Cnn- yn ) ) E KEN, y,z)char kf2, n> 3 (nI 3 points rn P2) h=a

€374I'

But actually , Ian-"E In foe n > 3 .

Harbourne's Coyectme is satisfied when :

• I general points in IPTBora'

-Harbourne) and p3CDumnicki)• I squarefree monomial ideal

In• RII f-pure/of dense F-pure type (G- Huneke)F

= ItCX ),X nxm generic matrix

• Rft E Veronese"J {. Rye ring of invariants of a linearly reductive group

RE f- regular ⇒ can replace h by h -Iwhen h=2

, get ICM = In for all n>I

stable Harbourne Conjecture Ichn - htt ) E In for noo .

-

(AK-htt )c

k

Question If I - I for some k,(kn-htt )

cn

does that imply I - I for all noo ?

NIE If yes, then Harbourne stable holds in ckarp .

theorem (G) If Ichk-h)E Ik for some k

then Ican - h) e In for all n⇒O

tote No known examples that fail the hypothesis .

Resurgence ( Boni - Harbourne)fCI ) = sup {F : I of Ib}

1-EfCI ) Eh

Note : If GCI ) ah, then Harborne stable holds.

why ?HIC > f CI ) ⇒ Ichn - C ) e In

In >C-h-g to

Assume h > 2 .

Question can gCI ) = he ?

I has expected resurgence if GCI ) ch .

theorem (G- Huneke-Mukundan )Crim) RLR/polynomial ring wth I homogeneousAssume I

" '= In : M ?

(eg , I = I pane of height d-1, or ideal of points )

If Ian-htt )em In for some n, then fetch .

Applications-

:

char O Ia '= In : m ?¥Homogeneous ideal generated in degree ache

a) I defining ctgtb.to)I'

EM Ia (Knodel- Schemed- Lanzhou)3) RII Goremtan

,Icn's In : mo