symbolic of · 2020. 11. 2. · symbolic powers and the (stable) containmentproblem...
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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