Medical, Pharma, Engineering, Science, Technology and Business

**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.

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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.

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

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 U_{l} (g)L^{d} 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 H_{l} ⊆ 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 U_{l} (g)L^{d} 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 U_{l} (g)L^{d} 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 T^{l}(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 . A resolution of the ideal sheaf of will give information on its syzygies. By literature of Maakestad [11] the first discriminant on the projective line 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 . 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.

In this section we study the jet bundle of any finite rank G-linearized locally free sheaf ε on the grassmannian as P_{l} -module, where 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 be closed subgroups. The following holds: There is a quotient morphism

(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 be the category of locally free- 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

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 be the ideal of the diagonal and let be the structure sheaf of the *n’th infinitesimal neigborhood of the diagonal.*

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

be the l’th jet bundle of ε.

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

(2.2.1)

*with G-linearization.*

*Proof*. By literature of Maakestad [4] sequence 2.2.1 is an exact sequence of locally free 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 modules with a G-linearization

Since p_{*} preserves G-linearization we get a morphism

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

Let g = Lie(G) and h = Lie(H). Let H_{l} ⊆ H be a Levi subgroup of H. It follows H_{l} is a maximal linearly reductive subgroup of H. The group H_{l} 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*

(2.3.1)

*of L-modules.*

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

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

Hence the study as H_{l}-module is reduced to the study of ε(x)* and Sym^{l}(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 = (m, m + n) is the grassmannian of m-planes in V. Let g = *Lie*(G) and p = *Lie*(P). Fix a basis e_{1}, .., e_{m} for W and e_{1}, .., e_{m}, e_{m+1}, .., e_{m+n} for V. It follows the K-rational points of P are matrices M on the form

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

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

**Proposition 2.4.** *There is a canonical isomorphism*

*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

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 *there is an isomorphism*

*of P _{l} -modules.*

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

There is an isomorphism of P-modules

hence the decomposition into irreducible components of the module as *P _{l}* -module may be done using the

Let λ − |i| denote λ is a partition of the integer i If λ = {λ_{1}, .., λ_{d}} is a partition of an integer l, let μ (λ) denote the following partition:

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

**Corollary 2.6.** *There is an isomorphism*

of SL(W) × SL(V/W)-*modules*.

*Proof*. By Corollary 2.5 there is an isomorphism

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

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

**Example 2.7.** Calculation of the cohomology group

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

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

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

since P_{semi} 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 we can use this construction to calculate the cohomology group

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

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

**Corollary 2.8.** *There is an isomorphism*

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

*Proof*. This follows from Corollary 2.5 and Proposition 5.1.

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 e_{1}, .., e_{m} and V have basis e_{1}, .., e_{m+n}. Let V * have basis x_{1}, .., x_{m+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 is the grassmannian of m-planes in V. Let Let L^{d} = Sym^{d}(∧mW). There is an inclusion of P-modules L^{d} ⊆ Sym^{d}(∧mV). Since K has characteristic zero there is an inclusion of G-modules

Let g = Lie(G) and p = Lie(P). Let U(g) be the universal enveloping algebra og g and let U_{l} (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

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

of P-modules. The highest weight vector for is the line L^{d} = Sym^{d}(∧mW). Let ann(L^{d}) ⊆ U(g) be the left annihilator ideal of L^{d}. It is the ideal generated by elements x ∈ U (g) with the property x(L^{d}) = 0. Let *ann _{l}* (L

and an exact sequence of P-modules

for all l ≥ 1. The G-module is the * generalized Verma module* corresponding to the P-module defined by L

**Definition 3.1.** *be the canonical filtration for *

**Lemma 3.2.** Assume y ∈ g and with x_{i} ∈ g. The following holds:

*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:

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

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

**Proposition 3.3.**

*The sub Lie algebra p _{L} ⊆ p is a sub P-module of p.* (3.3.1)

*There is an exact sequence of P-modules*

(3.3.2)

*and p/pL is the trivial P-module.*

*The following holds:*

(3.3.3)

*There is a filtration of P-modules*

(3.3.4)

*with quotients*

*for* 1 ≤ i ≤ k.

*Assume dim _{k}(W) = 1 and let W = L. There is an exact sequence of P-modules*

(3.3.5)

*giving an isomorphism of P-modules*

*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

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

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

and *tr*(aAa−1) + *tr*(aa−1A) = *tr*(A) = 0 hence g(x) ∈p_{L} and 3.3.1 is proved.

We prove 3.3.2: By 3.3.1 it follows p_{L} ⊆ p is a sub P-module. One checks p/p_{L} 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

*dim _{K}(g) = (m + n)^{2} − 1 = n^{2} + 2mn + m^{2}−1*

and

*dim _{K}(pL) = m^{2} + mn + n^{2} − 2*

it follows *dim _{K}*(g/p

We prove 3.3.4: Since p/p_{L} is a trivial P-module there are isomorphisms of P-modules

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

defined by

The injection j gives rise to an injection

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

0 → p/p_{L} → g/p_{L} →g/p→ 0

gives rise to a filtration of P-modules

with quotients

There is an isomorphism

and claim 3.3.4 is proved.

We prove 3.3.5: Let V = K{e_{0}, .., e_{n}} and L = W = e_{0}. It follows P ⊆ G = SL(V) is the group whose K-rational points are the following:

with 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:

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

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

We get two P-modules: p_{L} ⊆ p and L^{i} = Sym^{i} (∧^{m}W) ⊆ Sym^{i} (∧^{m}V). We get for all 1 ≤ k ≤ d a P-module

There is an injection of P-modules

defined by

There are natural embeddings of P-modules

and

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

**Proposition 3.4.** Let *The following formula holds:*

Proof. we prove the result by induction on k. Assume k=1 and let x(L^{d})∈U_{1}(g)L^{d}. It follows and the claim holds for k =1. Assume the result is true for k. Hence

with Assume

We get

Let and Such elements exist since as P-module. Let

it follows Moreover

where . The Proposition is proved.

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

*of P-modules.*

*Proof.* There are embeddings of P-modules

and

Recall from studies of Maakestad [1] it follows where *dim _{k}*(V) = n+1. Assume. It follows from Proposition 3.4

where

Since

it follows Hence we get an inclusion of P-modules

Since

the Theorem follows.

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

*of P-modules.*

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

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

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 be the structure morphism. Let Since Sym^{1}(V*) is a finite dimensional SL(V)-module it follows it is a free -module with an SL(V)-linearization. It follows π*Sym^{1}(V*) is a locally free -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*

*of locally free -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

(3.7.1)

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

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

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

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

*of locally free sheaves.*

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

Let and consider the P-modules

and

**Proposition 3.9.** *There is an isomorphism*

*of P-modules.*

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

It follows

hence there is an inclusion and the Proposition is proved.

**Corollary 3.10.** *There is an isomorphism*

*of P-modules.*

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

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

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

for all i, j.

**Example 3.11.** Canonical filtration for the grassmannian

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

and

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

where L_{x} is the line spanned by the following vector x:

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

and

Let be the vector space spanned by the vectors x_{1}, x_{2}, x_{4}, x_{4} and x. It follows The vector space V has a basus e_{1}, e_{2}, e_{3} and e_{4}. The vector space W has basis e_{1}, e_{2}. It follows ∧^{2}W has a basis given by e_{1}∧ e_{2} = e[12] and ∧^{2}V 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:

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

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

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

and

There is no equality

of P-modules as submodules of Sym^{d}(∧^{m}V) in general as Example 3.11 shows.

Since and 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 for all l. There is similarly a canonical line

Hence the two P-modules U_{l} (g)L^{d} and look similar.

In general the SL(V)-module Sym^{d}(∧^{m}V) decompose

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

with the property that the morphism

induce an isomorphism

of P-modules. In general the SL(V)-module Sym^{d}(∧^{m}V) has lots of automorphisms. When m = 2 and *dim _{k}*(V) = 4 it follows by Corollary 5.4 there is for every d ≥ 1 an equality

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

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

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

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

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 where 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:

by

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

by

**Lemma 4.2.** *The following holds for all* p ≥ 1: d^{p} ο d^{p−1} = 0.

*Proof*. We get

and the claim of the Lemma follows.

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

called the *Koszul complex of the map φ*

**Example 4.3.** *The Koszul complex of a regular sequence.*

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

φ : E → F

by

Let It follows

looks as follows:

Hence the complex equals the Koszul complex of the regular sequence . It is an exact complex since 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 be a map of locally free -modules. Let

be defined locally by

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

by

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

is called the *Koszul complex of* φ. Here

**Example 4.5.** *Koszul complexes and local complete intersections.*

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

and

Assume the image

has

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

is a resolution of the ideal I_{i} since I_{i} is generated by a regular sequence. The complex is isomorphic to the Koszul complex on the regular sequence It follows the global complex

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

Since the ideal Ii is generated by a regular sequence of length r it follows *dim*(A_{i} / I_{i}) = dim(A_{i}) − 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 on the projective line.*

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

where and consider the following diagram

There is a sequence of locally free -modules

and let be the composed map

(12)

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

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

defined by

We get the map

defined by

The composed map

is the map

Let and let. Let

Restrict the map 4.6.1 to the open set U_{i1}

We get the following two maps of modules:

defined by

We get the map

defined by

The composed map

is the map

It follows the ideal sheaf is generated by

on U_{i0} and by

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

*be an arbitrary degree d polynomial with a _{d} ≠ 0. Let f^{(i)}(t) denote the formal derivative with respect to t. It follows*

*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

and the claim of the Lemma follows.

**Lemma 4.8.** *The sequence {z _{l}, .., z_{0}} is a regular sequence in K[u_{i}, t]. The sequence {w_{l}, .., w_{0}} is a regular sequence in K[u_{i}, s].*

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

which is a domain, hence w_{l−1} is a non zero divisor in A[t] / w_{l}. By induction it follows z_{l}, .., z_{0} is a regular sequence in A[t]. Assume i ≤ l. It follows the sequence z_{l}, .., z_{i+1} is a regular sequence in A[t]. We see from Lemma 4.7 z_{l} is non zero in

and is a domain. It follows z_{i} is a non zero divisor in A[t]/(z_{l}, .., z_{i+1}). It follows z_{l}, .., z_{0} is a regular sequence in A[t] and the claim follows. A similar argument proves w_{l}, .., w_{0} is a regular sequence in A[s] and the Lemma is proved.

One may prove using similar methods for any permutation σ ∈ S_{l+1} the sequences

and

are regular sequences.

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

The morphism

gives by Example 4.3 rise to a Koszul complex

of locally free sheaves of

**Definition 4.9.** Let the complex

(4.9.1)

be the *incidence complex of *

Since the ideal sheaf of 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

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

Let V = K{e_{0}, e_{1}} and and consider the diagram

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

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

By the equivariant projection formula for higher direct images we get

Let

It follows

We get

We get the following Theorem:

**Theorem 4.10.** *The following holds:*

*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

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 on more general grassmannians and flag varieties. The hope is we may be able to construct resolutions of the ideal sheaf of 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 is a determinantal scheme for any d ≥ 2 on the projective line Assume is a G-linearized linebundle, G a semi simple linear algebraic group and P a parabolic subgroup. If one can prove D 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.

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 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 P_{semi}-modules where P_{semi} is the semi-simplification of P. It follows P_{semi} 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 P_{semi} module. We may apply the theory of highest weights since P_{semi} = 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*

(5.1.1)

*Proof*. Recall the canonical isomorphism from Lemma 2.4

of *P*-modules. It follows

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 be the Plucker embedding and let be tautological line bundle on G / P and let It follows from the Borel-Weil-Bott Theorem [16] is an irreducible SL(V)-module. Let V have basis e_{1}, e_{2}, e_{3}, e4 and let ∧^{2}V have basis e_{ij} for 1 ≤ i ≤ j ≤ 4, with e_{ij} = e_{i} ∧ e_{j}. Consider the element f ∈ Sym^{2}(∧^{2}V) where

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

is the defining equation for as closed subscheme of

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

(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

Dualize this sequence to get the split exact sequence

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

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

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 *where l = k if d = 2k or d = 2k + 1. It follows*

as SL(V)-module.

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

and the Corollary is proved.

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

*Proof*. This follows from Proposition 5.4 and the Borel-Weil-Bott theorem (BWB). From the BWB theorem it follows 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 Sym^{d} (∧^{2}V) is a multiplicity free SL(V)- module for all d ≥ 1. This is not true in general for Sym^{d}(∧^{m}K^{m+n}) when m, n > 2.

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

where 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.

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

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

**Proposition 6.1.** *Let U = K ^{m}. There is a canonical map of SL(V)- modules*

*defined by*

*Here e _{1}, .., e_{m} is a basis for U and x_{1}, .., x_{m} is a basis for U*.*

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

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

may be interpreted as a polynomial of degree m in the elements hence it is an element of Sym^{m}(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

where

and

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 e_{1}, .., e_{m} and V have basis e_{1}, .., e_{m}, f_{1}, .., f_{n}. Let W_{1}={e_{m}}, W_{2} = {e_{m}, e_{m−1}}, and

It follows we get a filtration

of the vector space W. Let

and let We get a surjection

V/W → V_{i}

for i = 1, .. , n−1. It follows *dimW _{i}* =

x_{i} : W_{i} → V_{i}

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

where

and

The i’th wedge product

may be viewed as an element in

via Proposition 6.1.

**Proposition 6.2.** *The following formula holds:*

*for all* g∈B. Here 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 with are all highest weight vectors for the module

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

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