Non-simple vector bundles 153
or h X, HomL , M. Just to fix the notation we assume h
X, HomL , M = 0. Every non- trivial pointwise nilpotent subalgebra B of H
X, EndE has nil-exponent two and dimension at most 1 + h
X, HomM, L. For any integer v with 0 ≤ v ≤ h X, HomM, L and for
every linesr subspace V of H X, HomM, L with dimV = v there is a pointwise nilpotent
subalgebra B of H X, EndE and the isomorphism class of B as abstract K-algebra depends
only from v, not the choice of V and are isomorphic to the algebra AV just described in the indecomposable case. A byproduct of the discussion just given is that E is A-stable if and only
if A ∼ = M
2×2
K and E ∼
= L ⊕ L. E
XAMPLE
4. Fix an integer a ≥ 2 and two vector bundles B, D on X such that h X,
HomB, D ≥ a − 1. Fix a linear subspace V of H X, HomB, D with dimV = a − 1 and
let DV := KI d ⊕ V be the unitary K-algebra obtained taking the trivial multiplication on V , i.e. such that uw = 0 for all u, w ∈ V . Notice that DV is commutative. Consider an extension
1 0 → B → E → D → 0
of D by B. There is a unique injection h : DV → H X, EndE of unitary K-algebras
obtained sending the element v ∈ V ⊂ DV into the endomorphism f
v
: E → E obtained as composition of the surjection E → D given by 1, the map v : D → B and the inclusion
B → E given by 1. P
ROPOSITION
3. Assume charK 6= 2. Let A be a commutative pointwise-nilpotent al-
gebra with nil-exponent two and E , h an A-sheaf. Set a := dim A. Then there exist vector bundles B, D and a linear subspace V of H
X, HomB, D with dimV = a − 1 such that, with the notation of Example 4, E fits in an exact sequence 1, A ∼
= DV and h is obtained as in Example 4, up to the identification of A with DV .
Proof. Take a general h ∈ h A and let λ be its eigenvalue. Set u = f − λI d, B
′
= Keru and D
′
= E B
′
. Since a ≥ 2, f ∈ KI d and hence u 6= 0. Thus D
′
6= {0}. Since Imu ⊆ E , B
′
is saturated in E . Hence D
′
is a vector bundle. Since u
2
= 0, B
′
6= {0}. There is a non-empty Zariski open subset W of A such that for every m ∈ W , calling λ
m
the eigenvalue associated to m, we have rankKerm − λ
m
I d = rankB
′
and degKerm − λ
m
I d = degB
′
. Set w =
m − λ
m
I d. Since u − w
2
= 0 and u
2
= w
2
= 0, we have uw + wu = 0. Since
A is commutative and charK 6= 2 we obtain uw = wu = 0. Since u
2
= w
2
= 0 we obtain Imu ⊆ Keru ∩ Kerw and Imw ⊆ Keru ∩ Kerw. Vary m in W and call B
the saturation of the union T of all subsheaves Imw
1
+ · · · + Imw
x
, x ≥ 1, and w
i
∈ W and nilpotent for every i . T is a coherent subsheaf of Keru because the set of all such sums
Imw
1
+ · · · + Imw
x
is directed and we may use [3], 0.12. Set D := E B. Thus we have an exact sequence 1. We just proved that B is contained in Kerw for all nilpotent w coming from
some f ∈ W . Since W is dense in h A, we have B ⊆ Kerw for every nilpotent w ∈ h A, i.e. every f ∈ h A is obtained composing the surjection E → D given by 1 with a map D ∈ B
and then with the inclusion of B in E given by 1. Hence h A ∼ = DV for some V.
References
[1] M
ARUYAMA
M., On boundedness of families of torsion free sheaves, J. Math. Kyoto Univ.
21 1981, 673–701. [2] S
ESHADRI
C. S., Fibr´es vectoriels sur les courbes alg´ebriques, Asterisque 96, Soc. Mat.
France 1982.
154 E. Ballico
[3] S
IU
Y. T.
AND
T
RAUTMANN
G., Gap-sheaves and extensions of coherent analytic sub-
sheaves, Lect. Notes in Math. 172, Springer-Verlag, Berlin - Heidelberg - New York 1971.
AMS Subject Classification: 14H60
Edoardo BALLICO Dipartimento di Matematica
Universit`a of Trento 38050 Povo TN, ITALIA
e-mail: ballicoscience.unitn.it
Rend. Sem. Mat. Univ. Pol. Torino Vol. 59, 2 2001
Liaison and Rel. Top.
C. Bocci - G. Dalzotto GORENSTEIN POINTS IN
P
3
Abstract. After the structure theorem of Buchsbaum and Eisenbud [1] on Goren- stein ideals of codimension 3, much progress was made in this area from the al-
gebraic point of view; in particular some characterizations of these ideals using h−vectors Stanley [9] and minimal resolutions Diesel [3] were given. On the
other hand, the Liaison theory gives some tools to exploit, but, at the same time, it requires one to find, from the geometric point of view, new Gorenstein schemes.
The works of Geramita-Migliore [5] and Migliore-Nagel [6] present some con- structions for Gorenstein schemes of codimension 3; in particular they deal with
points in
P
3
. Starting from the work of Migliore and Nagel, we study their constructions and we
give a new construction for points in P
3
: given a specific subset of a plane com- plete intersection, we add a “suitable” set of points on a line not in the plane and
we obtain an aG zeroscheme that is not complete intersection. We emphasize the interesting fact that, by this method, we are able to “visualize” where these points
live.
1. Introduction
