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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Jet Bundles on Projective Space II

Maakestad H*

Tempelveien 112, 3475 Sætre i Hurum, Bærum, Norway

Corresponding Author:
Maakestad H
Tempelveien 112, 3475 Sætre i Hurum
Bærum, Norway
Tel: +4798457343
E-mail: [email protected]

Received date: October 21, 2015; Accepted date: November 24, 2015; Published date: December 01, 2015

Citation: Maakestad H (2015) Jet Bundles on Projective Space II. J Generalized Lie Theory Appl S2:001. doi:10.4172/generalized-theory-applications.S2-001

Copyright: © 2015 Maakestad H. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

In previous papers the structure of the jet bundle as P-module has been studied using different techniques. In this paper we use techniques from algebraic groups, sheaf theory, generliazed Verma modules, canonical filtrations of irreducible SL(V)-modules and annihilator ideals of highest weight vectors to study the canonical filtration Ul (g)Ld of the irreducible SL(V)-module H0 (X, ïX(d))* where X = ï(m, m + n). We study Ul (g)Ld using results from previous papers on the subject and recover a well known classification of the structure of the jet bundle ïl (ï(d)) on projective space ï(V*) as P-module. As a consequence we prove formulas on the splitting type of the jet bundle on projective space as abstract locally free sheaf. We also classify the P-module of the first order jet bundle ïX1 (ïX (d)) for any d ≥ 1. We study the incidence complex for the line bundle ï(d) on the projective line and show it is a resolution of the ideal sheaf of I l (ï(d)) - the incidence scheme of ï(d). The aim of the study is to apply it to the study of syzygies of discriminants of linear systems on projective space and grassmannians.

Keywords

Algebraic group; Jet bundle; Grassmannian; P-module; Generalized verma module; Higher direct image; Annihilator ideal; Canonical filtration; Discriminant; Koszul complex; Regular sequence; Resolution

Introduction

In a series of papers of Maakestad [1-4], the structure of the jet bundle as P-module has been studied using different techniques. In this paper we continue this study using techniques from algebraic groups, sheaf theory, generalized Verma modules, canonical filtrations of irreducible SL(V)-modules and annihilator ideals of highest weight vectors and study the canonical filtration of the SL(V)-module where is the grassmannian of m-planes in an m + n-dimensional vector space. Using results obtained in studies of Maakestad [1] we classify Ul (g)Ld and as a corollary we recover a well known result on the structure of the jet bundle as P-module. As a consequence we get well known formulas on the splitting type of the jet bundle on projective space as abstract locally free sheaf. We also classify the P-module of the first order jet bundle d on any grassmannian (Corollary 3.10).

In the first section of the paper we study the jet bundle of any locally free G-linearized sheaf ε on any quotient G / H. Here G is an affine algebraic group of finite type over an algebraically closed field K of characteristic zero and H ⊆ G is a closed subgroup. There is an equivalence of categories between the category of finite dimensional H-modules and the category of finite rank locally free -modules with a G-linearization. The main result of this section is Theorem 2.3 where we give a classification of the Hl -modules structure of the fiber where Hl ⊆ H is a Levi subgroup. Here x G / H is the distinguished K-rational point defined by the identity e ∈ G. We also study the structure of as Hl-module where X = (m, m + n) is the grassmannian of m-planes in an m + n-dimensional vector space (Corollary 2.5 and 2.8).

In the second section we study the canonical filtration Ul (g)Ld for the irreducible SL(V)-module Here We prove in Theorem 3.5 there is an isomorphism

of P-modules when is projective n -space. As a result we recover in Corollary 3.6 the structure of the fiber as P-module. This result was proved in another paper [5] using different techniques. We also recover in Corollary 3.8 a known formula on the structure of the jet bundle on projective space as abstract locally free sheaf [2,6-10].

In the third section we study the incidence complex

of the line bundle on the projective line. Using Koszul complexes and general properties of jet bundles we prove it is a locally free resolution of the ideal sheaf of - the incidence scheme of .

In Appendix A and B we study SL(V)-modules, automorphisms of SL(V)-modules and give an elementary proof of the Cauchy formula.

Hence the paper initiates a general study of the canonical filtration Ul (g)Ld for any line bundle with d ≥ 1 on any grassmannian (m, m + n) as P-module. In Section 3 we show some of the complications arising in this study by giving explicit examples.

The study of the jet bundle of a line bundle on the grassmannian is motivated partly by its relationship with the discriminant of the line bundle . There is by studies of Maakestad [11] for all 1 ≤ l < d an exact sequence of locally free -modules

giving rise to a diagram of maps of schemes

Where is the restriction of the projection map and i, j are closed immersions. By definition is the schematic image of via π. The K-rational points of are pairs of K-rational points (s, x) with the property that Tl(x)(s) = 0 in . The scheme is the incidence scheme of the l’th Taylor morphism

The map π is a surjective generically finite morphism between irreducible schemes. There is by literature of Maakestad [11] a Koszul complex of locally free sheaves on
(1.0.1)

which is a resolution of the ideal sheaf of when it is locally generated by a regular sequence. The complex 1 might give information on a resolution of the ideal sheaf of equation. A resolution of the ideal sheaf of equation will give information on its syzygies. By literature of Maakestad [11] the first discriminant equation on the projective line image is the classical discriminant of degree d polynomials, hence it is a determinantal scheme. By the results of Lascoux [12], we get an approach to the study of the syzygies of equation. Hence we get two approaches to the study of syzygies of discriminants of line bundles on projective space and grassmannians: One using Taylor maps, incidence schemes, jet bundles and generalized Verma modules. Another one using determinantal schemes.

Jet Bundles on Quotients

In this section we study the jet bundle of any finite rank G-linearized locally free sheaf ε on the grassmannian image as Pl -module, where image is a maximal linearly reductive subgroup.

Let K be an algebraically closed field of characteristic zero and let V be a K-vector space of dimension n. Let image be closed subgroups. The following holds: There is a quotient morphism

image(2.0.2)

and G / H is a smooth quasi projective scheme of finite type over K. Moreover

H ⊆ G is parabolic if and only if G / H is projective. (2.0.3)

For a proof refer to literature of Jantzen [13]. Let X = G / H and let image be the category of locally freeimage- modules with a G-linearization. Let mod(H) be the category of finite dimensional H-modules. It follows from Jantzen [13], there is an exact equivalence of categories

image

Let Y = G / H × G / H and p, q : Y → G / H be the canonical projection maps. The scheme G / H is smooth and separated over Spec(K) hence the diagonal morphism

Δ : G / H → Y

is a closed immersion of schemes. Let image be the ideal of the diagonal and let image be the structure sheaf of the n’th infinitesimal neigborhood of the diagonal.

Definition 2.1. Let ε be a locally free finite rank image module. Let

image

be the l’th jet bundle of ε.

Proposition 2.2. There is for all l ≥ 1 an exact sequence of locally free image modules

image (2.2.1)

with G-linearization.

Proof. By literature of Maakestad [4] sequence 2.2.1 is an exact sequence of locally free image modules. The scheme Y is equipped with the diagonal G-action. It follows p* and q* preserve G-linearizations. We get a diagram of exact sequences of image modules with a G-linearization

image

Since p* preserves G-linearization we get a morphism

image

preserving the G-linearization, and the Proposition is proved.

Let g = Lie(G) and h = Lie(H). Let Hl ⊆ H be a Levi subgroup of H. It follows Hl is a maximal linearly reductive subgroup of H. The group Hl is not unique but all such groups are conjugate under automorphisms of H. Let x ∈ G / H be the K-rational point defined by the identity e ∈ G.

Theorem 2.3. There is for all l ≥ 1 an isomorphism

image (2.3.1)

of L-modules.

Proof. Dualize the sequence 2.2.1 and take the fiber at x to get the exact sequence

image

of H-modules (and Hl-modules). This sequence splits since Hl is linearly reductive and the Theorem follows by induction on l.

Hence the study image as Hl-module is reduced to the study of ε(x)* and Syml(g/h).

Let W ⊆ V be K-vector spaces of dimension m and m + n and let G = SL(V) and P ⊆ G the subgroup fixing W. It follows G / P = image(m, m + n) is the grassmannian of m-planes in V. Let g = Lie(G) and p = Lie(P). Fix a basis e1, .., em for W and e1, .., em, em+1, .., em+n for V. It follows the K-rational points of P are matrices M on the form

image

where det(A)det(B) = 1, A an m × m-matrix and B an n × n-matrix. Let image be the subgroup defined as follows: The K-rational points of Pl are matrices M on the form

image

where det(A)det(B) = 1 and similarly A an m × m-matrix and B an n × n-matrix. It follows Pl is a Levi subgroup of P, hence it is a maximal linearly reductive subgroup.

Proposition 2.4. There is a canonical isomorphism

image

of P-modules.

Proof. By definition g = sl(V), hence φ ∈ g is a map

φ : V → V

with tr(φ) = 0. Let i : W → V be the inclusion map and p : V → V / W the projection map. Define the following map:

J' : g→Hom(W, V /W)

by

j′(φ) = p ο φ i.

It follows j(p) = 0 hence we get a well defined map

j : g/p → Hom(W, V / W)

defined by

image

One checks g/p and Hom(W, V/W) are P-modules and j a morphism of P-modules. It is an isomorphism and the Proposition follows.

Corollary 2.5. On image there is an isomorphism

image

of Pl -modules.

Proof. The proof follows from Theorem 2.3 and Proposition 2.4.

There is an isomorphism of P-modules

image

hence the decomposition into irreducible components of the module image as Pl -module may be done using the Cauchy formula (Appendix B).

Let λ − |i| denote λ is a partition of the integer i If λ = {λ1, .., λd} is a partition of an integer l, let μ (λ) denote the following partition:

image

Let for any partition λ of an integer l and any vector space image denote the Schur-Weyl module of λ.

Corollary 2.6. There is an isomorphism

image

of SL(W) × SL(V/W)-modules.

Proof. By Corollary 2.5 there is an isomorphism

image

of Pl-modules and SL(W) × SL(V/W)-modules, since SL(W) × SL(V/W) ⊆ Pl is a closed subgroup. Since

image

the result follows from the Cauchy formula (Appendix B or [14]).

Example 2.7. Calculation of the cohomology group image

In the following we use the notation introduced in litertature of Jantzen [13]. Let Psemi = SL(m) × SL(n) ⊆ P be the semi simplification of P. We get a vector bundle

image

Let X = G / P and Y = G / Psemi Given any finite dimensional P-module W, let image denote its corresponding image-module. Let Wsemi denote the restriction of W to Psemi. By the results of Perkinson [13] it follows there is an isomorphism

image

of locally free sheaves. This will help calculating the higher cohomology group

image

since Psemi is semi simple and π is a locally trivial fibration. If W is the P-module corresponding to the dual of the j’th exterior power of the jet bundle image we can use this construction to calculate the cohomology group

image

Such a calculation will be by the results of Maakestad [11], Example 5.12 give information on resolutions of the ideal sheaf of Dl(image (d)) since the push down of the Koszul complex 1.0.1 is the locally trivial sheaf

image

To describe the locally trivial sheaf image for all i, j we need to calculate the dimension image and this calculation may be done using the approach indicated above.

image

Corollary 2.8. There is an isomorphism

image

of SL(2) × SL(2)-modules. Here image if i = 2n + 1.

Proof. This follows from Corollary 2.5 and Proposition 5.1.

On Canonical Filtrations and Jet Bundles on Projective Space

In this section we study the canonical filtration for the dual of the SL(V)-module of global sections of an invertible sheaf on the grassmannian. We classify the canonical filtration on projective space and as a result recover known formulas on the splitting type of the jet bundle as abstract locally free sheaf.

Let W ⊆ V be vector spaces over K of dimension m and m + n. Let W have basis e1, .., em and V have basis e1, .., em+n. Let V * have basis x1, .., xm+n. Let G = SL(V) and P ⊆ G the parabolic subgroup of elements fixing W. It follows there is a quotient morphism

π : G → G / P

and image is the grassmannian of m-planes in V. Let image Let Ld = Symd(∧mW). There is an inclusion of P-modules Ld ⊆ Symd(∧mV). Since K has characteristic zero there is an inclusion of G-modules

image

Let g = Lie(G) and p = Lie(P). Let U(g) be the universal enveloping algebra og g and let Ul (g) be the l’th term to its canonical filtration.

By the Corollary 3.11 in studies of Maakestad [15] there is for all 1 ≤ l ≤ d an exact sequence of P-modules

image

Since the grassmannian is projectively normal in the Plucker embedding we get an inclusion

image

of P-modules. The highest weight vector for image is the line Ld = Symd(∧mW). Let ann(Ld) ⊆ U(g) be the left annihilator ideal of Ld. It is the ideal generated by elements x ∈ U (g) with the property x(Ld) = 0. Let annl (Ld) be its canonical filtration. We get an exact sequence of G-modules

image

and an exact sequence of P-modules

image

for all l ≥ 1. The G-module image is the generalized Verma module corresponding to the P-module defined by Ld = Symd(∧mV). There is an inclusion of P-modules

image

Definition 3.1.image be the canonical filtration for image

Lemma 3.2. Assume y ∈ g and image with xi ∈ g. The following holds:

image
image

Proof. The proof is by induction.

The Lie algebra p is the sub Lie algebra of g = sl(V) given by matrices M of the following type:

image

where A is an m × m-matrix, B and n × n-matrix and tr(A) + tr(B) = 0. Let pL be the sub Lie algebra of p consisting of matrices M ∈ p of the following type:

image

where tr(A) + tr(B) = 0.

Proposition 3.3.

The sub Lie algebra pL ⊆ p is a sub P-module of p. (3.3.1)

There is an exact sequence of P-modules

image (3.3.2)

and p/pL is the trivial P-module.

The following holds:

image (3.3.3)

There is a filtration of P-modules

image (3.3.4)

with quotients

image

for 1 ≤ i ≤ k.

Assume dimk(W) = 1 and let W = L. There is an exact sequence of P-modules

image (3.3.5)

giving an isomorphism of P-modules image

Proof. We prove 3.3.1: In the following A, a are square matrices of size m and b, B square matrices of size n. The K -rational points of the group P are matrices g on the form

image

where det(A)det(B) = 1. Assume x ∈ p is the following element:

image

with tr(a) + tr(b) = 0. It follows g(x) = gxg−1 has tr(gxg−1) = tr(gg−1x) = tr(x) = 0 hence gxg−1 ∈ p and p is a P-module. Assume x ∈pL ie tr(a) = tr(b) = 0. It follows

image

and tr(aAa−1) + tr(aa−1A) = tr(A) = 0 hence g(x) ∈pL and 3.3.1 is proved.

We prove 3.3.2: By 3.3.1 it follows pL ⊆ p is a sub P-module. One checks p/pL is a trivial P-module. We clearly get an exact sequence of P-modules and 3.3.2 is proved.

We prove 3.3.3: Since

dimK(g) = (m + n)2 − 1 = n2 + 2mn + m2−1

and

dimK(pL) = m2 + mn + n2 − 2

it follows dimK(g/pL) = mn + 1. It follows

image

We prove 3.3.4: Since p/pL is a trivial P-module there are isomorphisms of P-modules

image

for all 1 ≤ i ≤ k. We get an injection

image

defined by

image

The injection j gives rise to an injection

image

of P-modules for all 1 ≤ i ≤ k. The exact sequence

0 → p/pL → g/pL →g/p→ 0

gives rise to a filtration of P-modules

image

with quotients

image
image

There is an isomorphism

image

and claim 3.3.4 is proved.

We prove 3.3.5: Let V = K{e0, .., en} and L = W = e0. It follows P ⊆ G = SL(V) is the group whose K-rational points are the following:

image

with image Also B is an n × n-matrix with coefficients in K. By definition the maps in the sequence are maps of P-modules. It follows p = Lie(P) is the Lie algebra whose elements x are matrices on the following form:

image

where B is any n × n-matrix with coefficients in K. The sub Lie algebra pL ⊆ p is the Lie algebra of matrixes x ∈ p on the following form:

image

where B is any n × n-matrix with tr(B) = 0. Let xi ∈ g be the following element: Let the first column vector of xi be the vector ei and let the rest of the entries be such that tr(xi) = 0. It follows image L and xi(e0) = ei hence the vertical map is surjective. One easily checks the sequence is exact and 3.3.5 is proved.

We get two P-modules: pL ⊆ p and Li = Symi (∧mW) ⊆ Symi (∧mV). We get for all 1 ≤ k ≤ d a P-module

image

There is an injection of P-modules

image

defined by

image

There are natural embeddings of P-modules

image

and

image

Assume in the following m = 1 and L = W. It follows imageimage is projective n-space.

Proposition 3.4. Let image The following formula holds:

image
image

Proof. we prove the result by induction on k. Assume k=1 and let x(Ld)∈U1(g)Ld. It follows image and the claim holds for k =1. Assume the result is true for k. Hence

image

with image Assume

image

We get

image

Let image andimage Such elements exist since image as P-module. Let

image

it follows image Moreover

image

where image. The Proposition is proved.

Theorem 3.5. There is for all 1 ≤ l ≤ d an isomorphism

image

of P-modules.

Proof. There are embeddings of P-modules

image

and

image

Recall from studies of Maakestad [1] it follows image where dimk(V) = n+1. Assumeimage. It follows from Proposition 3.4

image

where

image

Since

image

it follows image Hence we get an inclusion of P-modules image

Since

image

the Theorem follows.

Corollary 3.6. There is for all 1≤ l ≤ d an isomorphism

image

of P-modules.

Proof. There is by studies of Maakestad [1], Theorem 3.10 an isomorphism

image

of P-modules. From this isomorphism and Theorem 3.5 the Corollary follows since

image

as P-modules.

Note: Corollary 3.6 is proved in literature of Maakestad [5] Theorem 2.4 using more elementary techniques.

Let Y = Spec(K) and image be the structure morphism. Let image Since Sym1(V*) is a finite dimensional SL(V)-module it follows it is a free image-module with an SL(V)-linearization. It follows π*Sym1(V*) is a locally free image-module with an SL(V)-linearization since π* preserves the SL(V)-linearization.

Proposition 3.7. There is for all 1≤ l ≤ d an isomorphism

image

of locally free image-modules with an SL(V)-linearization.

Proof. Let P ⊆ SL(V) be the subgroup fixing the line L ∈ V There is an exact equivalence of categories

image (3.7.1)

The P-module corresponding to image isimage By the equivalence 3.7.1 and Corollary 3.6 we get an isomorphism

image

of locally free sheaves with SL(V)-linearization and the Proposition is proved.

We get a formula for the splitting type of image on projective space:

Corollary 3.8. There is for all 1 ≤ l ≤ d an isomorphism

image

of locally free sheaves.

Proof. The P-modules Sym1(V*) corresponds to the free image-module image The Corollary now follows from Proposition 3.7.

Let image and consider the P-modules

image

and

image

Proposition 3.9. There is an isomorphism

image

of P-modules.

Proof. Pick an element image It follows image hence there is an inclusion

image
image It follows

image

hence there is an inclusion image and the Proposition is proved.

Corollary 3.10. There is an isomorphism

image

of P-modules.

Proof. There is by studies of Maakestad [1], Theorem 3.10 an isomorphism

image

of P-modules. The Corollary follows from this fact and Proposition 5.1.

Note: By studies of Maakestad [11], Example 5.12 there is a double complex

image

of sheaves on image where image and image This double complex might give rise to a resolution of the ideal sheaf of the l’th discriminant image of the line bundleimage By the literature of Maakestad, Theorem 5.2 it follows knowledge on the P-module structure of image gives information on the SL(V)-module structure of the higher cohomology groups image for all i ≥ 0. This again gives information on the dimension image We get a description of the locally free sheaf

image

for all i, j.

Example 3.11. Canonical filtration for the grassmannian image

Consider the example where m = n = 2 and image We get two inclusions

image

and

image

We may choose a basis for p ⊆ g on the following form:

image

where Lx is the line spanned by the following vector x:

image

Let n ⊆ g be the sub Lie algebra spanned by the following vectors:

image
image

and

image

Let image be the vector space spanned by the vectors x1, x2, x4, x4 and x. It follows image The vector space V has a basus e1, e2, e3 and e4. The vector space W has basis e1, e2. It follows ∧2W has a basis given by e1∧ e2 = e[12] and ∧2V has basis given by e[12], e[13], e[14], e[23], e[24], e[34]. By definition L = e[12]. We get the following calculation:

image

A basis for the P-module image are the following vectors:

image

Let a = d(d−1). A basis for the P-module imageare the following vectors:

image
image

In the case where W ⊆ V have dimensions m and m + n we get embeddings of P-modules

image

and

image

There is no equality

image

of P-modules as submodules of Symd(∧mV) in general as Example 3.11 shows.

Since image and image by Theorem 3.5 and Proposition 3.3 are isomorphic when m = 1 and 1 ≤ l ≤ d, have the same dimension over K and both have natural filtrations of P-modules we may conjecture they are isomorphic as P-modules for all m,n ≥ 1. Note: There is a canonical line image for all l. There is similarly a canonical line

image

Hence the two P-modules Ul (g)Ld and image look similar.

In general the SL(V)-module Symd(∧mV) decompose

image

where image are irreducible SL(V)-modules and ai ≥ 1 are integers (Proposition 5.4 for the situation of image. One may ask if there is a non-trivial automorphism

image

with the property that the morphism

image

induce an isomorphism

image

of P-modules. In general the SL(V)-module Symd(∧mV) has lots of automorphisms. When m = 2 and dimk(V) = 4 it follows by Corollary 5.4 there is for every d ≥ 1 an equality

image

where l = k if d = 2k or d = 2k + 1. For m = n = 2 the SL(V)-module Symd(∧mV) is by Proposition 5.4 multiplicity free. The module Symd(∧mKm+n) is not multiplicity free in general when m, n > 2.

Jet Bundles and Incidence Complexes on the Projective Line

In this section we construct a resolution by locally free sheaves of the ideal sheaf of the l’th incidence scheme image Here image is an invertible sheaf on the projective line image and image There is on image a morphism image of locally free sheaves

image

Its zero scheme image is the l’th incidence scheme of image The Koszul complex of the morphismimage.

image

- called the incidence complex of image is a resolution of the ideal sheaf of image This follows from the fact that the ideal sheaf of I image is locally generated by a regular sequence. We also calculate the higher direct images of the terms

image

appearing in the incidence complex.

The aim of the construction is to use it to construct a resolution of the ideal sheaf of the discriminant image where image is a line bundle on projective space or a grassmannian.

Example 4.1. The Koszul complex of a map of locally free modules.

Let A be an arbitrary commutative ring with unit and let φ : E → F be a map A-modules.

Define the following map:

image

by

image

Let I A be the image of d1. We let Iφ be the ideal of φ. Define the following map

image

by

image

Lemma 4.2. The following holds for all p ≥ 1: dp ο dp−1 = 0.

Proof. We get

image
image

and the claim of the Lemma follows.

Assume E, F are locally free of finite rank and let image We get a complex of locally free A-modules

image

called the Koszul complex of the map φ

Example 4.3. The Koszul complex of a regular sequence.

Let image be a regular sequence of elements in A and let E = Ae be the free A-module on the element e. Let image be a free rank n module on image Define

φ : E → F

by

image

Let image It follows

image

looks as follows:

image

Hence the complex image equals the Koszul complex image of the regular sequence image. It is an exact complex since image is a regular sequence.

Example 4.4. The Koszul complex of a morphism of locally free sheaves.

The construction of the differential in the Koszul complex of a map of modules is intrinsic, hence we may generalize to morphisms of locally free sheaves. Let Y be an arbitrary scheme and let image be a map of locally free image -modules. Let

image

be defined locally by

image

Let image be the ideal sheaf defined by d 1. Since image is quasi coherent sheaf of ideals it follows the ideal sheaf image corresponds to a subscheme image - the zero scheme of φ. Let U ⊆ Y be an open subset and define the following map:

image

by

image

This gives a well defined map of locally free sheaves since we have not chosen a basis for the module image to give a definition. By Lemma 4.2 it follows image for all for all p ≥ 1 hence we get a complex of locally free sheaves. The sequence of maps of locally free sheaves

image

is called the Koszul complex of φ. Here image

Example 4.5. Koszul complexes and local complete intersections.

Assume image is a map of locally free image-modules where image is a line bundle. Let Z(φ) ⊆ Y be the subscheme defined by φ - the zero scheme of φ. Let r = rk image. Choose an open affine cover Ui of Y where image and image trivialize, i.e

image

and

image
image Assume the image

image

has

image

where image is a regular sequence. Let image It follows from Example 4.3 the Koszul complex

image

is a resolution of the ideal Ii since Ii is generated by a regular sequence. The complex image is isomorphic to the Koszul complex image on the regular sequenceimage It follows the global complex

image

is a resolution of the ideal sheaf imagesince it is locally isomorphic to the Koszul complex image for all i.

Since the ideal Ii is generated by a regular sequence of length r it follows dim(Ai / Ii) = dim(Ai) − r. If Y is irreducible of dimension d it follows Z (φ) ⊆ Y is a local complete intersection of dimension d − r.

Example 4.6. The incidence complex of image on the projective line.

Let image where K is a field of characteristic zero and let image ∈ Pic image = Z be a line bundle where d ∈ Z. Let

image

where image and consider the following diagram

image

There is a sequence of locally free image-modules

image

and let image be the composed map

image (12)

It follows by studies of Maakestad [11], the zero scheme Z(φimage) equals the incidence scheme image of the line bundle image By definitionimage whereimage It has an open cover on the following form: image where we let image
image

Restrict the map 4.6.1 to the open set image We get the following two maps of modules:

image

defined by

image

We get the map

image

defined by

image

The composed map

image

is the map

image
image

Let image and letimage. Let

image

Restrict the map 4.6.1 to the open set Ui1

We get the following two maps of modules:

image
image

defined by

image

We get the map

image

defined by

image

The composed map

image

is the map

image
image

It follows the ideal sheaf image is generated by

image

on Ui0 and by

image
image

Lemma 4.7. Assume B is a commutative ring of characteristic zero and let

image

be an arbitrary degree d polynomial with ad ≠ 0. Let f(i)(t) denote the formal derivative with respect to t. It follows

image

Proof. The proof is by induction. It is clearly true for l = 1. Assume it is true for l > 1. Consider k = l + 1. We get

image

and the claim of the Lemma follows.

Lemma 4.8. The sequence {zl, .., z0} is a regular sequence in K[ui, t]. The sequence {wl, .., w0} is a regular sequence in K[ui, s].

Proof. Let image andimage Assume l < i and consider the sequence image Since A[t] is a domain it follows zl is a non zero divisor in A[t]. We see from Lemma 4.7

image

which is a domain, hence wl−1 is a non zero divisor in A[t] / wl. By induction it follows zl, .., z0 is a regular sequence in A[t]. Assume i ≤ l. It follows the sequence zl, .., zi+1 is a regular sequence in A[t]. We see from Lemma 4.7 zl is non zero in

image

and image is a domain. It follows zi is a non zero divisor in A[t]/(zl, .., zi+1). It follows zl, .., z0 is a regular sequence in A[t] and the claim follows. A similar argument proves wl, .., w0 is a regular sequence in A[s] and the Lemma is proved.

One may prove using similar methods for any permutation σ ∈ Sl+1 the sequences

image

and

image

are regular sequences.

It follows the ideal sheaf image is locally generated by a regular sequence.

The morphism

image

gives by Example 4.3 rise to a Koszul complex

image

of locally free sheaves of image

Definition 4.9. Let the complex

image (4.9.1)

image

be the incidence complex of image

Since the ideal sheaf of image by the discussion above is locally generated by a regular sequence it follows from Example 4.3 the complex 4.9.1 is a resolution.

In framework of Maakestad [5], Theorem 5.10 one calculates the higer direct images

image

for all i, j. We get the following calculations:

Let V = K{e0, e1} and image and consider the diagram

image

By the results of this paper it follows there is an isomorphism

image

a sheaves with an SL(V)-linearization. We get

image

By the equivariant projection formula for higher direct images we get

image

Let

image

It follows

image

We get

image
image

We get the following Theorem:

Theorem 4.10. The following holds:

image
image

Proof. The proof follows from the calculation of the equivariant cohomology of line bundles on projective space [13].

Hence we have complete control on the sheaf

image

on the projective line and projective space for all i, j. Using the techniques introduced in this paper one may describe resolutions of incidence schemes image on more general grassmannians and flag varieties. The hope is we may be able to construct resolutions of the ideal sheaf of image using indicence resolutions in a more general situation.

Note: In literature of Lascoux [12] resolutions of ideal sheaves of determinantal schemes are studied and much is known on such resolutions. In studies of Maakestad [11] it is proved image is a determinantal scheme for any d ≥ 2 on the projective line image Assumeimage is a G-linearized linebundle, G a semi simple linear algebraic group and P a parabolic subgroup. If one can prove D image is a determinantal scheme we get two approaches to the study of resolutions of ideal sheaves of discriminants: One using jet bundles and incidence schemes, another one using determinantal schemes.

Appendix A: Automorphisms of Representations

Let W ⊆ V be vectorspaces of dimension two and four over the field K. Consider the subgroup P ⊆ G = SL(V) where P is the parabolic subgroup of elements fixing W. It follows image is a principal P-bundle. Let g = Lie(G) and p = Lie(P) be the Lie algebras of G and P. In this section we study the decomposition into irreducibles and automorphisms of some G-modules. We also study some Psemi-modules where Psemi is the semi-simplification of P. It follows Psemi equals SL(2) × SL(2). Since p ⊆ g is a P-sub module it follows the quotient g/p is a P-module hence a Psemi module. We may apply the theory of highest weights since Psemi = SL(2) × SL(2) is a semi simple algebraic group.

Proposition 5.1. The following hold: There is an isomorphism of SL(2) × SL(2)-modules

image (5.1.1)

image

Proof. Recall the canonical isomorphism from Lemma 2.4

image

of P-modules. It follows

image

and its decomposition into irreducible SL(2) × SL(2)-modules can be done using well known formulas [14]. Alternatively one may compute its highest weight vectors and highest weights explicitly using the construction from Section 5.

Let image be the Plucker embedding and letimage be tautological line bundle on G / P and letimageimage It follows from the Borel-Weil-Bott Theorem [16] image is an irreducible SL(V)-module. Let V have basis e1, e2, e3, e4 and let ∧2V have basis eij for 1 ≤ i ≤ j ≤ 4, with eij = ei ∧ ej. Consider the element f ∈ Sym2(∧2V) where

image

One checks f is a highest weight vector for SL(V) with highest weight 0, hence it defines the unique trivial character of SL(V). Its dual

image

is the defining equation for image as closed subscheme of image

Proposition 5.2. The following hold: there is an isomorphism of SL(V)-modules

image (5.2.1)

where l = k if d = 2k or d = 2k + 1.

Proof. The result is proved using the theory of highest weights. There is a split exact sequence of SL(V)-modules

image

Dualize this sequence to get the split exact sequence

image

where image Since f is the trivial character it follows there is an isomorphism

image

of SL(V)-modules. By the Borel-Weil-Bott Theorem it follows Qd is an irreducible SL(V)-module. If d = 2k we get by induction the equality

image

and the claim of the Proposition is proved in the case where d = 2k. The claim when d = 2k + 1 follows by a similar argument and the Proposition is proved.

Corollary 5.3. Let image where l = k if d = 2k or d = 2k + 1. It follows

image

as SL(V)-module.

Proof. We get by Proposition 5.4 isomorphisms of SL(V)-modules

image
image

and the Corollary is proved.

Corollary 5.4. There is for every d ≥ 1 an equality

image
image

Proof. This follows from Proposition 5.4 and the Borel-Weil-Bott theorem (BWB). From the BWB theorem it follows image is an irreducible SL(V)-module for all d ≥ 1. From this and Proposition 5.4 the claim of the Corollary follows.

Hence the SL(V)-module Symd (∧2V) is a multiplicity free SL(V)- module for all d ≥ 1. This is not true in general for Symd(∧mKm+n) when m, n > 2.

In general if image andimage are two Schur-Weyl modules [14] there is a decomposition

image

where image is an irreducible SL(V)-module for all i. It is an open problem to calculate this decomposition for two arbitrary partitions λ and μ.

Appendix B: The Cauchy Formula

We include in this section an elementary discussion of the Cauchy formula using multilinear algebra. Let W ⊆ V be vector spaces of dimension m and m + n over K and let P ⊆ SL(V) be the subgroup fixing W. Let g = Lie(G) and p = Lie(P). There is a canonical isomorphism

image

of P-modules, hence the elements of g/p may be interpreted as linear maps. The symmetric power image (Hom(W, V / W)) is a P-module hence a Psemi = SL(m) × SL(n)-module and we want to give an explicit construction of its highest weight vectors as Psemi- module.

Proposition 6.1. Let U = Km. There is a canonical map of SL(V)- modules

image

defined by

image

Here e1, .., em is a basis for U and x1, .., xm is a basis for U*.

Proof. The proof is left to the reader as an exercise.

Note: in Proposition 6.1 the element image is an element of image = Hom(U,U). Hence the determinant

image

may be interpreted as a polynomial of degree m in the elements image hence it is an element of Symm(Hom(U,U)).

Let B ⊆ SL(m,K) × SL(n,K) ⊆ SL(V) = SL(m + n,K) be the following subgroup: B consists of matrices with determinant one of the form

image

where

image

and

image

Let T be a B-module and v ∈ T a vector with the property that for all x ∈ B it follows

xv = λ(x)v

where λ ∈ (Hom(B,K*) is a character of B. It follows v is a highest weight vector for T as SL(m, K) × SL(n, K)-module. The group B ⊆ SL(V) defines filtrations of W and V/W as follows: Let W have basis e1, .., em and V have basis e1, .., em, f1, .., fn. Let W1={em}, W2 = {em, em−1}, and

image

It follows we get a filtration

image

of the vector space W. Let

image

and let image We get a surjection

V/W → Vi

for i = 1, .. , n−1. It follows dimWi = dimVi = di for all i. Let x : W → V/ W be a linear map of vector spaces. We get an induced map

xi : Wi → Vi

wich is a square di matrix for all i. Let g∈B be the element

image

where

image

and

image

The i’th wedge product

image

may be viewed as an element in

image

via Proposition 6.1.

Proposition 6.2. The following formula holds:

image

for all g∈B. Here image is a character λ ∈ Hom(B, K*).

Proof. The proof is left to the reader as an exercise.

Hence the i’th determinant | xi | ∈ Symi(Hom(W,V/W)) is a highest weight vector for the SL(m) × SL(n)-module Symi(Hom(W,V/W)). By the results of studies Brion [17-22], it follows the vectors image with image are all highest weight vectors for the module

image

Acknowledgements

The author thanks Michel Brion, Alexei Roudakov and an anonymous referee for comments on the contents of this paper.

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