Inverse systems, fat points and
the Weak Lefschetz Property
Alexandra Seceleanu
University of Illinois at Urbana-Champaign
AMS National Meeting in New Orleans
January 2011
Alexandra Seceleanu (UIUC)
Inverse systems, fat points and the WLP
1 / 13
The Weak Lefschetz Property
Let I ⊆ S = K[x1 , . . . , xr ] be an ideal such that A = S/I is Artinian.
Definition
A graded Artinian algebra A has the Weak Lefschetz Property (WLP)
if there is an element ` ∈ S1 such that for all degrees j, the map
·`
µ` : Aj −→ Aj+1 is either injective or surjective (equivalently µ` has
maximum rank as a K-vector space homomorphism).
Alexandra Seceleanu (UIUC)
Inverse systems, fat points and the WLP
2 / 13
The Weak Lefschetz Property
Let I ⊆ S = K[x1 , . . . , xr ] be an ideal such that A = S/I is Artinian.
Definition
A graded Artinian algebra A has the Weak Lefschetz Property (WLP)
if there is an element ` ∈ S1 such that for all degrees j, the map
·`
µ` : Aj −→ Aj+1 is either injective or surjective (equivalently µ` has
maximum rank as a K-vector space homomorphism).
The set of linear forms ` with this property is Zariski open in S1 .
We assume henceforth that ` ∈ S1 is generic.
We assume also that char(K) = 0.
Alexandra Seceleanu (UIUC)
Inverse systems, fat points and the WLP
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Background
WLP is
known to hold for monomial Artinian complete intersections
(Stanley, 1980).
expected to hold for Artinian ideals generated by generic forms.
known to hold for Artinian ideals generated by generic forms in
K[x1 , x2 , x3 ] (Anick, 1986).
known to hold for any Artinian ideal generated by powers of linear
forms in K[x1 , x2 , x3 ] (Schenck - S., 2009).
known to hold for Artinian ideals generated by generic forms in
K[x1 , x2 , x3 , x4 ] (Migliore - Miró-Roig, 2003).
Alexandra Seceleanu (UIUC)
Inverse systems, fat points and the WLP
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Background
WLP is
known to hold for monomial Artinian complete intersections
(Stanley, 1980).
expected to hold for Artinian ideals generated by generic forms.
known to hold for Artinian ideals generated by generic forms in
K[x1 , x2 , x3 ] (Anick, 1986).
known to hold for any Artinian ideal generated by powers of linear
forms in K[x1 , x2 , x3 ] (Schenck - S., 2009).
known to hold for Artinian ideals generated by generic forms in
K[x1 , x2 , x3 , x4 ] (Migliore - Miró-Roig, 2003).
How about powers of linear forms in K[x1 , x2 , x3 , x4 ]?
Alexandra Seceleanu (UIUC)
Inverse systems, fat points and the WLP
3 / 13
Motivating Example: 5 points on a conic
Look ahead:
I = `31 , `32 , `33 , `34 , `35 ⊆ K[x1 , x2 , x3 , x4 ].
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Inverse systems, fat points and the WLP
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Motivating Example: 5 points on a conic
Look ahead:
I = `31 , `32 , `33 , `34 , `35 ⊆ K[x1 , x2 , x3 , x4 ].
The space of quartics in P2 passing through five double points is
nonempty =⇒ WLP fails for geometric reasons.
Alexandra Seceleanu (UIUC)
Inverse systems, fat points and the WLP
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First tool - (Macaulay) inverse systems
Let {p1 , . . . , pn } ⊆ Pr−1 be a set of distinct points defined by ideals
I(pi ) = ℘i ⊆ R = K[y1 , . . . , yr ]. A fat point ideal is an ideal of the form
F =
n
\
i
℘m
i ⊂ R.
i=1
Alexandra Seceleanu (UIUC)
Inverse systems, fat points and the WLP
5 / 13
First tool - (Macaulay) inverse systems
Let {p1 , . . . , pn } ⊆ Pr−1 be a set of distinct points defined by ideals
I(pi ) = ℘i ⊆ R = K[y1 , . . . , yr ]. A fat point ideal is an ideal of the form
F =
n
\
i
℘m
i ⊂ R.
i=1
Recall S = K[x1 , . . . , xr ] and define an action of R on S by partial
differentiation: yj · xi = ∂xi /∂xj .
Definition
The set of elements annihilated by the action of F is denoted F −1 and
called the (Macaulay) inverse system associated to the ideal F .
Alexandra Seceleanu (UIUC)
Inverse systems, fat points and the WLP
5 / 13
Linear forms come into play
Emsalem and Iarrobino proved that there is a close connection between
ideals generated by powers of linear forms and ideals of fat points.
Theorem (Emsalem and Iarrobino)
Let F be an ideal of fatpoints:
mn
1
F = ℘m
1 ∩ · · · ∩ ℘n ⊂ R.
Linear forms come into play
Emsalem and Iarrobino proved that there is a close connection between
ideals generated by powers of linear forms and ideals of fat points.
Theorem (Emsalem and Iarrobino)
Let F be an ideal of fatpoints:
mn
1
F = ℘m
1 ∩ · · · ∩ ℘n ⊂ R.
Then
(F
−1
)j =


Sj
for j < max{mi }

 j−m1 +1
n +1
Lp1
Sm1 −1 + · · · + Lpj−m
Smn −1
n
for j ≥ max{mi }
and
dimK (F −1 )j = dimK (R/F )j .
Second tool -the syzygy bundle
Harima-Migliore-Nagel-Watanabe have introduced the syzygy bundle as a
crucial tool in studying the WLP.
Definition
If I = hf1 , . . . , fn i is hx1 , . . . , xr i−primary, and deg(fi ) = di , then the
^ is a rank n − 1 bundle defined via
syzygy bundle S(I) = Syz(I)
0
/ Syz(I)
/
n
L
S(−di )
[f1 ,...,fn ]
/ S −→ S/I
/ 0.
i=1
Most importantly, H 1 (S(I)(j)) = Aj .
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Inverse systems, fat points and the WLP
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WLP and the syzygy bundle
The long exact sequence in cohomology given by the restriction of the
syzygy bundle to a hyperplane L defined by the linear form l yields:
0
GF
@A
GF
@A
/ H 0 (S(I)(j))
_______/ Aj
___/ H 2 (S(I)(j))
Alexandra Seceleanu (UIUC)
·`
/ H 0 (S(I)(j + 1))
/ H 0 (S(I)|L (j + 1))
ED
BC
/ Aj+1
/ H 1 (S(I)|L (j + 1))
BC
ED
/ ....
Inverse systems, fat points and the WLP
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Divisors on blowups
mn
1
Consider the fat points ideal F = ℘m
1 ∩ · · · ∩ ℘n ⊂ R.
On the blowup X of Pr−1 at the points p1 , . . . , pn , let
Ei be the class of the exceptional divisor over the point pi
E0 be the pullback of a hyperplane on Pr−1
The divisor
Dj = jE0 −
n
X
(j − mi + 1)Ei .
i=1
describes the global sections of the syzygy bundle
h0 (S(I)(j)) = h1 (Dj )
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Inverse systems, fat points and the WLP
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Special divisors in P2
Definition
A linear system of degree d through a set of fat points ℘1 , . . . , ℘n with
multiplicities m1 , · · · , mn in P
P2 is special
if its dimension excedes the
d+2
expected dimension 2 − ni=1 mi2+1 − 1.
E.g. The linear system of quartics through 5 double points has
negative expected dimension, but its actual dimension is 1.
By Riemann-Roch, the space of global sections (or H 0 cohomology)
is larger than expected iff the H 1 cohomology 6= 0.
Definition
We say D = dE0 −
Pn
i=1 mi Ei
is special if h0 (D) and h1 (D) are positive.
Motivating Example revisited
Let I = `31 , `32 , `33 , `34 , `35 ⊂ S = K[x1 , x2 , x3 , x4 ] and let A = S/I.
The Hilbert function of A is:
j
dimK Aj
Alexandra Seceleanu (UIUC)
0
1
1
4
2
10
3
15
4
15
5
6
Inverse systems, fat points and the WLP
6
0
...
...
11 / 13
Motivating Example revisited
Let I = `31 , `32 , `33 , `34 , `35 ⊂ S = K[x1 , x2 , x3 , x4 ] and let A = S/I.
The Hilbert function of A is:
j
dimK Aj
0
@A
GF
GF
@A
0
1
1
4
/ H 0 (S(I)(3)
________/ A3
·`
____/ H 2 (S(I)(3))
Alexandra Seceleanu (UIUC)
2
10
3
15
4
15
5
6
6
0
...
...
/ H 0 (S(I)(4))
/ H 0 (S(I)|L (4))
ED
BC
/ A4
/ H 1 (S(I)|L (4))
BC
ED
/ ....
Inverse systems, fat points and the WLP
11 / 13
Motivating Example revisited
Let I = `31 , `32 , `33 , `34 , `35 ⊂ S = K[x1 , x2 , x3 , x4 ] and let A = S/I.
The Hilbert function of A is:
j
dimK Aj
0
@A
GF
GF
@A
0
1
1
4
15
0
Alexandra Seceleanu (UIUC)
4
15
5
6
0
·`
____/ H 2 (S(I)(3))
3
15
/ H 0 (S(I)(4))
/ H 0 (S(I)(3)
________/ A3
2
10
15
/ A4
6
0
...
...
1
/ H 0 (S(I)|L (4))
1
/ H 1 (S(I)|L (4))
ED
BC
BC
ED
/ ....
Inverse systems, fat points and the WLP
11 / 13
Special divisors and (-1)-curves in P2
Conjecture (Segre-Harbourne-Gimigliano-Hirschowitz SHGH)
P
If D = dE0 − ni=1 mi Ei is a special divisor on a blowup of P2 , then there
exists a (−1)-curve E with E · D ≤ −2. ((−1)-curve means E · E = −1)
This conjecture is known to be true for n ≤ 8 points.
Special divisors and (-1)-curves in P2
Conjecture (Segre-Harbourne-Gimigliano-Hirschowitz SHGH)
P
If D = dE0 − ni=1 mi Ei is a special divisor on a blowup of P2 , then there
exists a (−1)-curve E with E · D ≤ −2. ((−1)-curve means E · E = −1)
This conjecture is known to be true for n ≤ 8 points.
Theorem (S.)
P
If E = dE0 − 8i=1 mi Ei is the divisor of a (−1)-curve on a blowup of P2
at n ≤ 8 points, then the coefficients are given by
d = 0, mi = (−1, 0, 0, 0, 0, 0, 0, 0)
d = 1, mi = (0, 0, 0, 0, 0, 0, 1, 1)
d = 2, mi = (0, 0, 0, 1, 1, 1, 1, 1)
d = 3, mi = (0, 1, 1, 1, 1, 1, 1, 2)
d = 4, mi = (1, 1, 1, 1, 1, 2, 2, 2)
d = 5, mi = (1, 1, 2, 2, 2, 2, 2, 2)
d = 6, mi = (2, 2, 2, 2, 2, 2, 2, 3)
Main results
Set Dj = jE0 −
n
P
(t + j − 1)Ei . Imposing that Dj · E ≤ −2, we obtain:
i=1
Theorem (Harbourne - Schenck- S.)
Let I = hl1t , . . . , lnt i ⊆ K[x1 , x2 , x3 , x4 ] = S with li ∈ S1 generic. If
n ∈ {5, 6, 7, 8}, then WLP fails, respectively, for t ≥ {3, 27, 140, 704}.
Alexandra Seceleanu (UIUC)
Inverse systems, fat points and the WLP
13 / 13
Main results
Set Dj = jE0 −
n
P
(t + j − 1)Ei . Imposing that Dj · E ≤ −2, we obtain:
i=1
Theorem (Harbourne - Schenck- S.)
Let I = hl1t , . . . , lnt i ⊆ K[x1 , x2 , x3 , x4 ] = S with li ∈ S1 generic. If
n ∈ {5, 6, 7, 8}, then WLP fails, respectively, for t ≥ {3, 27, 140, 704}.
Conjecture (Harbourne - Schenck- S.)
For I = hl1t , . . . , lnt i ⊆ K[x1 , . . . , xr ] = S with li ∈ S1 generic and
n ≥ r + 1 ≥ 5, WLP fails for all t 0.
Alexandra Seceleanu (UIUC)
Inverse systems, fat points and the WLP
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