A geometric approach to the Weak Lefschetz
Property for height four complete intersections
Special Session on Commutative Algebra and
Its Interactions with Algebraic Geometry
Dalhousie University
October 18–19, 2014
Juan C. Migliore
University of Notre Dame
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
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Goal:
Describe part of a long-standing project with Mats Boij, Rosa Marı́a
Miró-Roig and Uwe Nagel (started in May, 2011).
The Weak Lefschetz Property (WLP) is a seemingly very algebraic
notion which, nevertheless, has many connections to geometry. These
manifest themselves in different forms, some surprising.
We’ll look at one such connection, involving hyperplane sections of
certain smooth curves.
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
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Background
Let k be an infinite field of characteristic zero.
R = k [x1 , . . . , xr ], graded polynomial ring.
Let
A = R/I =
n
M
Ai
i=0
be a standard graded artinian k -algebra. Note that A is finite
dimensional over k.
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
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Let ` be a linear form. For each i, multiplication by ` induces a
homomorphism
×`
Ai −→ Ai+1 .
For a general choice of `, we “expect” maximal rank:
rk (×`) = min{dim Ai , dim Ai+1 }.
(It’s enough to find just one such `.)
Philosophy: something interesting should be going on if this is not
the case.
In our situation, it will have to do with the question of whether a certain
set of points imposes the “expected” number of conditions on forms of
a certain degree.
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
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Definition
A standard graded artinian k-algebra A has the Weak Lefschetz
Property (WLP) if, for a general linear form `, the homomorphism
(×`) : Ai → Ai+1
has maximal rank for all i (i.e. for each i it is either injective or
surjective). Any such linear form is a Lefschetz element.
We say that A has the Strong Lefschetz Property (SLP) if
(×`d ) : Ai → Ai+d
has maximal rank for all i, d (i.e. is either injective or surjective).
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
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This result got the whole area started:
Theorem (Stanley (1980), J. Watanabe (1987), Reid-Roberts-Roitman
(1991))
Let R = k[x1 , . . . , xr ], where k has characteristic zero. Let I be an
artinian monomial complete intersection, i.e.
I = hx1a1 , . . . , xrar i.
Then R/I has the SLP.
I In particular, R/I has the WLP.
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
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Note: Monomial complete intersections are a (very) special kind of
complete intersection.
As a consequence, if we fix a1 , . . . , ar then the general complete
intersection of type (a1 , . . . , ar ) has the WLP.
Big Question: Do all graded artinian complete intersections in
characteristic zero have the WLP?
Here, for the most part, is what is known.
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
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Fact 1: (HMNW 2003)
In 2 variables, not only do all complete intersections have the WLP, but
in fact everything has the WLP (at least in characteristic zero).
Fact 2: (HMNW 2003)
Let R = k[x, y , z], where char(k ) = 0. Let I = (F1 , F2 , F3 ) be a
complete intersection. Then R/I has the WLP.
The proof of Fact 2 uses the Grauert-Mülich theorem for rank two
vector bundles on P2 , which we apply to the syzygy bundle of I.
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
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Remarks.
1
After [HMNW], other authors have extended the use of the syzygy
bundle and Grauert-Mülich to study the WLP for other kids of
algebras, most notably Brenner and Kaid.
However, it does not seem to help much with complete
intersections of height ≥ 4.
2
SLP is still wide open for complete intersections in three variables
(always assuming characteristic zero). It’s one natural direction
from the previous theorem.
3
In positive characteristic p, many authors have studied WLP and
SLP, and their failure, especially for monomial complete
intersections.
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
9 / 24
There are two main parts of our paper.
In one, we introduce the non-Lefschetz locus of an artinian graded
algebra, complete with a scheme structure.
We study it in the case of general complete intersections, and
monomial complete intersections.
If there’s time, I’ll say more about it at the end of this talk.
In the other part we begin the study of WLP for arbitrary complete
intersections in k[x1 , x2 , x3 , x4 ].
This is the topic of this talk.
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
10 / 24
Overview:
We consider height 4 artinian complete intersections with
generators of the same degree, in characteristic zero.
We translate the WLP question into a question involving general
hyperplane sections of a certain smooth curve.
The first cohomology of the ideal sheaf of this curve (i.e. its
Hartshorne-Rao module) is the desired complete intersection.
We apply methods from Castelnuovo theory and Davis’s theorem
to the general hyperplane section of this curve, and get partial
results about whether all such complete intersections have the
WLP.
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
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Theorem
Let R = k[x1 , x2 , x3 , x4 ] where k has characteristic zero. Let
A = R/I = R/hF1 , F2 , F3 , F4 i
where I is a complete intersection and deg Fi = d for all i.
(a) If d = 2 or 3 then A has the WLP.
(b) Let ` be a general linear form. Then the multiplication maps
×` : [A]t−1 → [A]t
3d + 1
are injective for t <
.
2
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
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Notation and first comments:
R = k [x1 , x2 , x3 , x4 ]
(char k = 0).
I = hF1 , F2 , F3 , F4 i is a complete intersection with
deg Fi = d ≥ 2 for i = 1, 2, 3, 4.
A = R/I. We want to know if A has the WLP.
The socle degree of R/I is 4d − 4, an even integer.
Thus the Hilbert function of R/I has one “peak,” in degree 2d − 2.
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
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For example, if d = 4 then 2d − 2 = 6 and the Hilbert function is
deg
0
1
1
4
2
10
3
20
4
31
5
40
6
44
7
40
8
31
9
20
10
10
11
4
12
1
Easy fact: R/I has the WLP if and only if
×`
[R/I]2d−3 −→ [R/I]2d−2
is injective. (You don’t have to worry about any other degree.)
Remark: Our main result gives injectivity up to degree ∼ 32 d, while we
need degree ∼ 2d.
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
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Construction:
After a general choice of basis, set
IC1 = hF1 , F2 i and IC2 = hF3 , F4 i, so I = hF1 , F2 , F3 , F4 i = IC1 + IC2 .
C1 and C2 are smooth, irred. disjoint curves of degree d 2 by Bertini.
⇒ C = C1 ∪ C2 is a smooth curve in P3 of degree 2d 2 .
C1 and C2 are complete intersection (hence ACM) curves in P3 .
Since C1 and C2 are ACM curves in P3 , IC = IC1 · IC2 (Serre). So IC has
four minimal generators, all of degree 2d.
Q: What’s the deficiency (= Rao) module of C, denoted M(C)?
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
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A: From the sequence
0 → IC1 ∩ IC2 → IC1 ⊕ IC2 → IC1 + IC2 → 0
we sheafify and take cohomology, summing over all twists.
Since I = IC1 + IC2 is artinian and C1 , C2 are ACM, we obtain
0
IC
IC1 ⊕ IC2
R
H∗1 (IC )
0
I
0
and hence
Juan C. Migliore (Notre Dame)
0
M(C) := H∗1 (IC ) ∼
= R/I.
Geometric approach to WLP
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Some ingredients of the proof of the main theorem of this talk:
The general hyperplane section of C is the union, Z = Z1 ∪ Z2 , of
two sets of d 2 points, each with the uniform position property
(UPP).
“Symmetry:” The general hyperplane section can’t have any
property that “favors” Z1 over Z2 or vice versa.
Let ` be a general linear form defining a plane H, and let t be any
integer. We have a standard exact sequence of sheaves
×`
0 → IC (t − 1) −→ IC (t) → IC∩H|H (t) → 0
leading to a long exact sequence in cohomology:
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
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×`
×`
0 → [IC ]t−1 −→ [IC ]t → [IC∩H ]t → [R/I]t−1 −→ [R/I]t →
H 1 (IC∩H (t)) → H 2 (IC (t − 1)) → H 2 (IC (t)) → 0
where IC∩H is the ideal of C ∩ H viewed in H ∼
= P2 . One can show
I
dim[IC ]2d−1 = 0, dim[IC ]2d = 4 (as already noted).
I
h2 (IC (t)) = 0 for t ≥ 2d − 3.
Then setting t = 2d − 2 above, we get
×`
0 → [IC∩H ]2d−2 → [R/I]2d−3 −→ [R/I]2d−2 → H 1 (IC∩H (2d − 2)) → 0.
So the WLP depends entirely on the Hilbert function of C ∩ H.
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
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WLP ⇔ dim[IC∩H ]2d−2 = 0.
Specifically:
What can we say about the h-vector (= first difference of the Hilbert
function) for C ∩ H ⊂ P2 ?
Fact: The value of the h-vector in degree 2d − 1 is always d, whether
WLP holds or not. This translates to the “expected” h-vector for C ∩ H:
degree
0
1
1
2
2
3
...
...
2d − 3
2d − 2
2d − 2
2d − 1
↑ −
2d − 1
d
↑
2d
0
↑ +
The expected value in degree 2d is then forced to be 0 since
deg C = 2d 2 . (Simple calculation.) One shows there’s a balanced
see-saw effect if WLP fails.
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
19 / 24
Using
the UPP property for the two “halves” of C ∩ H for general H,
the mentioned symmetry property,
the famous theorem of Davis,
one can show:
The h-vector of C ∩ H is of decreasing type (and a bit more).
The main theorem follows from a very careful study of the possibilities
remaining after these are taken into account.
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
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Example
Say d = 2. The expected h-vector of C ∩ H is (1, 2, 3, 2).
The “2” in degree 3 is forced.
The only other possible h-vector would be (1, 2, 2, 2, 1). This can be
shown to be impossible.
Hence R/I has the WLP.
Example
If d = 3, we also show WLP holds.
If d = 4 then injectivity is forced by our theorem except in the very
middle.
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
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The other part of the paper
For an artinian complete intersection I ⊂ R = k[x1 , . . . , xn ], define the
non-Lefschetz locus, LI ⊂ Pn−1 , to be the locus of linear forms that fail
to be Lefschetz elements.
Note WLP ⇔ codim LI ≥ 1.
Assume the socle degree of R/I is e. Then R/I has the WLP if
and only if, for a general linear form `,
×` : [R/I]b e−1 c → [R/I]b e+1 c is injective.
2
2
The part of LI coming from any two consecutive components of
R/I has a scheme structure (not necessarily reduced) and in fact
is determinantal.
As a scheme, LI turns out to be determined by the single ×`
mentioned above. (This comes from the Gorenstein property.)
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
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The codimension of LI is a finer measure of how “general” linear forms
behave than simply the question of whether WLP holds.
There is an expected codimension for LI depending on the Hilbert
function of R/I.
We show that in several situations (in terms of number of
variables), a general complete intersection has non-Lefschetz
locus of expected codimension.
In contrast, we show that a monomial complete intersection
almost always has non-Lefschetz locus of codimension 1 (and we
describe it).
This highlights a key difference between the monomial case and
the general case, despite the fact that both have WLP and SLP.
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
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Thank you.
Juan C. Migliore (Notre Dame)
Geometric approach to WLP
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