Hyperplane sections of projective bundle associated to the tangent bundle of $\mathbb{P}^2.$

TL;DR

This paper provides a comprehensive classification of hyperplane sections of the projective bundle related to the tangent bundle of al P^2, and explores their deformations within a specific algebraic variety.

Contribution

It offers a complete description of hyperplane sections of the projective bundle over al P^2 and characterizes their deformations in the flag variety SL_3()/B.

Findings

Classified all hyperplane sections of the projective bundle over al P^2.

Described possible deformations of certain Schubert divisors within SL_3()/B.

Abstract

In this note we give a complete description of all the hyperplane section of the projective bundle associated to the tangent bundle of $\mathbb{P}^2$ under its natural embedding in $\mathbb{P}^7.$ As an application one obtains a description of all possible deformations, in $\text{SL_3(\mathbb{C}}/B,$ of the co-dimension one sub-scheme which is the union of two fundamental Schubert divisors.

Equations

$$\ell (Z):={\text{dim}}_{\mathbb{C}}{\mathcal{O}}_Z.$$

$$0\to {\mathcal{O}}_{{\mathbb{P}}^2}\to {\mathcal{O}}_{{\mathbb{P}}^2}(1{)}^3\to T_{{\mathbb{P}}^2}\to 0,$$

$$H^1({\mathcal{O}}_{{\mathbb{P}}^2})=0$$

$$\left(\begin{array}[]{ccc}X&Y&Z\\ f_{1}&f_{2}&f_{3}\end{array}\right)$$

$$X{f}_2-Y{f}_1,X{f}_3-Z{f}_1,Y{f}_3-Z{f}_2.$$

$${\mathrm{H}}^0({\mathcal{O}}_{{\mathbb{P}}^2}(1{)}^3)\to {\mathrm{H}}^0(T_{{\mathbb{P}}^2})\to 0$$

$$(\lambda -1)XY,(\mu -1)XZ,(\mu -\lambda )YZ.$$

$$V=\{(1,0,0),(0,1,0),(0,0,1)\}$$

$${\mathrm{H}}^0(X,E)\simeq {\mathrm{H}}^0(\mathbb{P}(E),{\mathcal{O}}_{\mathbb{P}(E)}(1)).$$

$$I={\cap }_{i=1}^r{Q}_i{\cap }_{i=1}^s{F}_i,$$

$$Z_i=V(F_i),1\le i\le s,$$

$$\mathbb{P}(E{|}_{V_1(s)})\cap {\pi }^{-1}(U)=W_1\cap {\pi }^{-1}(U),$$

$$V=V(f_{1}Y-f_{2}X,f_{1}Z-f_{3}X,f_{2}Z-f_{3}Y).$$

$${\mathcal{O}}_V={\mathcal{O}}_Q/m_Q\oplus {\mathcal{O}}_P/I,$$

$${\text{dim}}_{\mathbb{C}}{\mathcal{O}}_P/I=2.$$

$${\mathcal{O}}_V={\mathcal{O}}_P/I,$$

$${\text{dim}}_{\mathbb{C}}{\mathcal{O}}_P/I=3.$$

$$\text{Spec}(\mathbb{C}[X,Y]/(X,Y^3))$$

$$\text{Spec}(\mathbb{C}[X,Y]/(X^2,Y^2,XY)).$$

$$T_{{\mathbb{P}}^2}{|}_{\ell }={\mathcal{O}}_{\ell }(1)\oplus {\mathcal{O}}_{\ell }(2),$$

$$W_1\simeq W_2\simeq \mathbb{P}({\mathcal{O}}_{{\mathbb{P}}^1}\oplus {\mathcal{O}}_{{\mathbb{P}}^1}(1))$$

Full text

Keywords: Projective bundle; natural embedding; hyperplane section. fundamental Schubert divisors, deformations.

1. Introduction

Throughout we work over the field of complex numbers $\mathbb{C}.$ Let $X$ be a projective variety embedded in a projective space $\mathbb{P}^{n}.$ Recently several researchers are lead to study the hyperplane sections of $X$ in $\mathbb{P}^{n}$(see [BMSS],[CHMN], [GIV]). Given $X\subset\mathbb{P}^{n},$ an interesting and challenging problem is to classify the subschemes $H\cap X$ of $X$ for varying hyperplanes $H\subset\mathbb{P}^{n}.$ This problem is in general very difficult. For example, if we denote by $V(2,d)$ the image of $\mathbb{P}^{2}$ in its $d$-tuple embedding, then one has the description for all the hyperplane sections only for $d\leq 3.$ For $d=1,2$ it is easy and for $d=3,$ see [Ne]. In the case $d=2$ there are three distinct classes of subschemes and in the case $d=3$ there are eight distinct classes of subschemes are there.

Let $\mathbb{P}(T_{\mathbb{P}^{2}})$ be the projective bundle associated to the tangent bundle of $T_{\mathbb{P}^{2}}.$ This threefold is naturally embedded in $\mathbb{P}^{7}.$ The aim of this short note is to describe explicitly all possible types of hyperplane sections of this embedding. Existence of such a complete description is very rare.

We have the following:

Theorem 1.1**.**

A hyperplane section of $\mathbb{P}(T_{\mathbb{P}^{2}})$ in its natural embedding in ${\mathbb{P}}^{7}$ is either

i) an irreducible surface of degree six in ${\mathbb{P}}^{6}.$

or

ii) the union of two non-singular degree three surfaces.

If the hyperplane section is irreducible then it is either a non-singular Del Pezzo surface or a singular rational normal surface. Moreover, if it is a singular rational normal surface then it is non-singular except one point at which the singularity is either $A_{1}$ type (i.e., quadratic cone singularity with the local equation $Y^{2}+XZ=0$ ) or $A_{2}$ type (i.e.,singularity with the local equation $Y^{3}+XZ=0$).

If the hyperplane section is union of two non-singular degree three surfaces then their intersection is either one irreducible non-singular rational curve or a union of two non-singular rational curves intersecting at a point.

Theorem 1.1 gives us all possible deformations of the codimension one subscheme $H$ which is union of two fundamental co-dimension one Schubert varieties in the homogenous space ${\text{SL}_{3}(\mathbb{C})}/B,$ where $\text{SL}_{3}(\mathbb{C})$ is the group of all $3\times 3$ matrices of determinant one over the field of complex numbers and $B$ is the subgroup of upper triangular matrices (see, Remark 2.8).

2. Proof of the Theorem

Throughout this note we use standard notations, see for example [GH] or [Ha].

Definition 2.1**.**

For a subscheme $Z,$ of dimension zero, of the projective space $\mathbb{P}^{n},$ the ${\mathbb{C}}$ vector space dimension of the ${\mathbb{C}}$ algebra ${\mathcal{O}}_{Z}$ is called the length of the subscheme $Z$ and is denoted by $\ell(Z).$ i.e.,

[TABLE]

The proof of Theorem 1.1 depends on the following:

Theorem 2.2**.**

Let $s$ be a non-zero section of the tangent bundle $T_{\mathbb{P}^{2}}$ of ${\mathbb{P}^{2}}.$ Then the scheme $V$ of zeros of $s$ is of one of the following types:

a) $V$ is reduced and consists of three distinct points,

b) $V$ is non reduced and is a union of a reduced point and a non reduced subscheme of length two,

c) $V$ is non reduced subscheme of length three, supported on a single point,

d) $V$ is the union of a line and a point not on the line,

e) $V$ is a line with an embedded point on it.

Proof: Let $\{X,Y,Z\}$ be a basis of the space of sections of the line bundle ${\mathcal{O}}_{\mathbb{P}^{2}}(1).$ We consider $X,Y$ and $Z$ as the variables giving the homogeneous coordinates on $\mathbb{P}^{2}.$ On $\mathbb{P}^{2}$ we have the Euler sequence

[TABLE]

where the map ${\mathcal{O}}_{\mathbb{P}^{2}}\to{\mathcal{O}}_{\mathbb{P}^{2}}(1)^{3}$ is given by $1\mapsto(X,Y,Z).$ Any section of ${\mathcal{O}}_{\mathbb{P}^{2}}(1)^{3}$ gives a section of $T_{\mathbb{P}^{2}}$ by projection. Since the group

[TABLE]

every section of $T_{\mathbb{P}^{2}}$ is the image of a section of ${\mathcal{O}}_{\mathbb{P}^{2}}(1)^{3}.$ A section of ${\mathcal{O}}_{\mathbb{P}^{2}}(1)^{3}$ is an ordered triple $(f_{1},f_{2},f_{3})$ where $f_{i}\,(1\leq i\leq 3)$ are linear forms in the variables $X,Y,Z.$ Two sections $(f_{1},f_{2},f_{3})$ and $(g_{1},g_{2},g_{3})$ of ${\mathcal{O}}_{\mathbb{P}^{2}}(1)^{3}$ map to the same section of $T_{\mathbb{P}^{2}}$ if the difference of these two sections is a scalar multiple of $(X,Y,Z).$ Let $s$ be a non-zero section of $T_{\mathbb{P}^{2}}$ and $(f_{1},f_{2},f_{3})$ be a section of ${\mathcal{O}}_{\mathbb{P}^{2}}(1)^{3}$ which maps to $s$ under the surjection given by the Euler sequence (1). Clearly the scheme $V$ of zeros of $s$ is equal to the subscheme of $\mathbb{P}^{2}$ on which the two sections $(X,Y,Z)$ and $(f_{1},f_{2},f_{3})$ of ${\mathcal{O}}_{\mathbb{P}^{2}}(1)^{3}$ are dependent, namely the scheme defined by the vanishing of the two by two minors of the matrix

[TABLE]

i.e., the scheme $V$ defined by the common zeros of the polynomials

[TABLE]

Since the second Chern class $c_{2}({\mathcal{O}}_{\mathbb{P}^{2}}(1)^{3})=3[R],$ where $[R]$ is the cohomology class of a point $R\in\mathbb{P}^{2},$ we see that if the support of $V$ is a finite set, then it must be one of the types a), b) or c). Moreover as $s$ is non zero we see that $V$ is defined by at least two linearly independent quadrics and hence it cannot contain a conic as a subscheme. On the other hand if the restrictions of the sections $(f_{1},f_{2},f_{3})$ and $(X,Y,Z)$ to a line $\ell$ are linearly dependent, then the line $\ell$ is contained in $V.$ In this case we see that $V$ is of the type $d)$ or $e)$ as at least two of the quadrics in (2) are linearly independent. $\hfill{\Box}$

Corollary 2.3**.**

If $s$ is a non-zero holomorphic vector field on the projective plane $\mathbb{P}^{2}$ which vanishes on a positive dimensional subscheme then that subscheme is necessarily a line i.e., given by zeros of a homogenous polynomial of degree one in three variables. More over, if $s$ vanishes on a line then it can vanish at most one another point on $\mathbb{P}^{2}.$

Proof: Since a holomorphic section of the Tangent bundle is a holomorphic vector field corollary is an immediate consequence of 2.2. $\hfill{\Box}$

Example 2.4**.**

Here we give examples to show that all the types mentioned in Theorem 2.2 do occur. For $\lambda,\mu\in\mathbb{C},$ consider the section $(X,\lambda Y,\mu Z)$ of ${\mathcal{O}}_{\mathbb{P}^{2}}(1)^{3}$ and denote by $s$ the induced section of $T_{\mathbb{P}^{2}}$ under the surjection

[TABLE]

obtained by the exact sequence (1). Note that $\lambda=\mu=1$ if and only if $s$ corresponds to the zero section of $T_{\mathbb{P}^{2}}.$ The scheme $V$ of zeros of the section $s$ is given by the vanishing of the quadrics

[TABLE]

If $\lambda-1,\mu-1,(\mu-\lambda)$ are all non zero, then

[TABLE]

consists of three distinct points and hence is of type $a)$ of Theorem 2.2. If $\lambda=0$ and $\mu=1$ then $V$ is the union of the line $Y=0$ and the point $(0,1,0)$ and the scheme is of type $d)$ of Theorem 2.2.

If $s$ is the section defined by $(Y,0,0)$ then the zero scheme of $s$ is given by the vanishing of the quadrics $Y^{2}$ and $YZ$ which is of type $e)$ of Theorem 2.2.

If $s$ is the section defined by $(X,Z,0)$ then the zero scheme of $s$ is given by the vanishing of the quadrics $XZ-YX,XZ$ and $Z^{2}$ which is of type $b)$ of Theorem 2.2.

If $s$ is the section defined by $(Y,Z,0)$ then the zero scheme of $s$ is given by the vanishing of the quadrics $XZ-Y^{2},YZ$ and $Z^{2}$ which is of type $c)$ of Theorem 2.2. $\hfill{\Box}$

Let $E$ be a vector bundle of rank two on a smooth projective surface $X.$ Let $\mathbb{P}(E)$ be the projective bundle associated to the bundle $E$ and $\pi:\mathbb{P}(E)\to X$ be the natural projection. Let ${\mathcal{O}}_{\mathbb{P}(E)}(1)$ be the relative ample line bundle quotient of $\pi^{*}(E)$ and let $\phi$ be the natural isomorphism

[TABLE]

Lemma 2.5**.**

For a non zero section $\sigma$ of $E,$ let $V_{0}(\sigma)$ (resp. $V_{1}(\sigma)$) be the union of zero dimensional (resp. one dimensional) components of the subscheme of $X$ defined by zeros of $\sigma.$ Let $s\neq 0$ be a section of $E$ and $\phi(s)$ be the corresponding section of ${\mathcal{O}}_{\mathbb{P}(E)}(1).$ The scheme of zeros of a section $\phi(s)$ is equal to $W_{0}\cup W_{1},$ where $W_{0}$ (resp.$W_{1}$) is a subscheme of $\mathbb{P}(E)$ isomorphic to the blow-up of $X$ along $V_{0}(s)$ (resp. is isomorphic to ${\mathbb{P}(E|_{V_{1}(s)}}))$.

Proof: To prove the Lemma, first we observe:

i) If the section $s$ is non zero at a point $x$ the subspace generated by it defines a unique point in the projective line $\mathbb{P}(E_{x})$ at which the section $\phi(s)$ vanishes.

ii) Let $U$ be an affine open subset of $X$ such that $E|_{U}\simeq{\mathcal{O}}_{U}^{2}.$ Now $s|_{U}$ gives a section of ${\mathcal{O}}_{U}^{2},$ say $(f_{1},f_{2}),$ then the associated section $\phi(s)|_{U}$ of ${\mathcal{O}}_{{\mathbb{P}}({\mathcal{O}}_{U}^{2})}(1)$ is $f_{1}X_{1}+f_{2}X_{2},$ where $X_{1},X_{2}\in{\text{H}}^{0}({\mathcal{O}}_{{\mathbb{P}}^{1}}(1))$ is a basis. Thus it follows that the subscheme of vanishing of $\phi(s)|_{U}$ is the same as that of $f_{1}X_{1}+f_{2}X_{2}.$

If $I$ is the ideal generated by $f_{1},f_{2}$ in ${\rm H}^{0}({\mathcal{O}}_{U}),$ then

[TABLE]

is the primary decomposition of $I,$ where $Q_{i},1\leq i\leq r$ (resp. $F_{i},1\leq i\leq s$) are primary ideals corresponding to irreducible components of $V_{0}(s)\cap U$ (resp. $V_{1}(s)\cap U$). Since $U$ is non-singular, the components of $V_{1}(s)$ are locally principal divisors. That is, the subschemes

[TABLE]

of $U$ are locally principal. Hence both $f_{1}$ and $f_{2}$ vanish along the closed sub-scheme $Z_{i}\subset V_{1}(s)\cap U,1\leq i\leq s,$ and we see that

[TABLE]

where $\pi:\mathbb{P}(E)\to X$ is the projection map. By i) we see that the blow-up of $U$ along the subscheme $V_{0}(s)\cap U$ is equal to $W_{0}\cap\pi^{-1}(U).$

Since $X$ is covered by open sets of the form $U$ in ii) the Lemma follows immediately.

Lemma 2.6**.**

Let $s$ be a non-zero section of the tangent bundle $T_{\mathbb{P}^{2}}$ of ${\mathbb{P}^{2}}.$ Assume that the support of the scheme $V(\subset{\mathbb{P}^{2}})$ of zeros of $s$ is at most two points. Then either

i) $V$ is a union of a reduced scheme supported at a point and a non reduced subscheme $V_{1}$ of length two with ${\mathcal{O}}_{V_{1}}$ isomorphic to ${\mathbb{C}}[X,Y]/{(X,Y^{2})},$

or

ii) $V$ is a non reduced scheme of length three supported at a point and ${\mathcal{O}}_{V}$ isomorphic to ${\mathbb{C}}[X,Y]/{(X,Y^{3})}.$

Proof: Notations as in the proof of Theorem 2.2. If $(f_{1},f_{2},f_{3})$ is a section of ${\mathcal{O}}_{\mathbb{P}^{2}}(1)^{3}$ that map to the section $s$ of $T_{\mathbb{P}^{2}}$ then

[TABLE]

The assumption on the support of the subscheme $V,$ implies that $V$ has to be either of type b) or c) of the Theorem 2.2. Observe that the cohomology class of the zero dimensional scheme $V$ is $3[R],$ where $[R]\in H^{4}(\mathbb{P}^{2},{\mathbb{Z}})$ is the class of a point $R\in\mathbb{P}^{2}.$ Thus length of ${\mathbb{C}}$ algebra ${\mathcal{O}}_{V}$ is three (i.e., $\text{dim}_{{\mathbb{C}}}{{\mathcal{O}}_{V}}=3$). If $V$ is of the type b) of Theorem 2.2, then

[TABLE]

where $Q,P\in\mathbb{P}^{2}$ are two distinct points and $m_{Q}$ is the maximal ideal of $Q$ and $I$ is an ideal contained in the maximal ideal $m_{P}$ of $Q$ satisfying

[TABLE]

It is easy to see that any subschemes of length two supported on single point of $\mathbb{P}^{2}$ is isomorphic to $\text{Spec}({\mathbb{C}}[X,Y]/{(X,Y^{2})}),$ we conclude that if $V$ is of type b) of the Theorem 2.2 then it corresponds to i) of the Lemma.

If $V$ is of the type c) of Theorem 2.2, then

[TABLE]

where $P\in\mathbb{P}^{2}$ is a point and $I$ is an ideal contained in the maximal ideal $m_{P}$ of $P$ satisfying

[TABLE]

Any subscheme whose support is a single point of $\mathbb{P}^{2}$ and is of length three is isomorphic to either

[TABLE]

or

[TABLE]

Claim: If $V$ is of the form c) of Theorem 2.2 then it corresponds to ii) of the Lemma. To prove the claim, with out loss of generality, we assume that $P=(0,0,1).$ Then the subscheme $V$ is contained in the affine open set $U=\{Z\neq 0\}$ of $\mathbb{P}^{2}.$ and $V=V(g_{1}Y-g_{2}X,g_{1}-g_{3}X,g_{2}-g_{3}Y)$ where $g_{i},(1\leq i\leq 3)$ are homogenous polynomials in $X,Y$ of degree less or equal to one and the ideal $I=(g_{1}Y-g_{2}X,g_{1}-g_{3}X,g_{2}-g_{3}Y).$ By using assumptions that $P=(0,0,1)$ and $V$ is of the form c) of Theorem 2.2 we conclude ${\mathcal{O}}_{V}\simeq{\mathbb{C}}[X,Y]/{(X,Y^{3})}.$ This proves the claim.

This completes the proof of the Lemma. $\hfill{\Box}$

Lemma 2.7**.**

Let $s$ be a non-zero section of the tangent bundle $T_{\mathbb{P}^{2}}$ of ${\mathbb{P}^{2}},$ and $\phi(s)$ be the corresponding section of the line bundle ${\mathcal{O}}_{\mathbb{P}(T_{\mathbb{P}^{2}})}(1).$ Assume that the support of the scheme $V(\subset{\mathbb{P}^{2}})$ of zeros of $s$ is at most two points.

i) If $V$ is a union of a reduced point $Q$ and a non reduced scheme of length two supported at point $P\in\mathbb{P}^{2},$ then the scheme $W(\subset\mathbb{P}(T_{\mathbb{P}^{2}}))$ of zeros of $\phi(s)$ is a rational normal surface with exactly one singular point which is of the type $A_{1}$ (see, [R] §4.2)

ii) If $V$ is a non reduced scheme of length three supported at a point $P$ then the scheme $W(\subset\mathbb{P}(T_{\mathbb{P}^{2}}))$ of zeros of $\phi(s)$ is a rational normal surface with exactly one singular point which is of the type $A_{2}$ (see, [R] §4.2)

Proof: From the Lemma 2.5 and its proof we deduce the following:

If $V$ is as in i) then the scheme $W$ is isomorphic to blow-up of ${\mathbb{P}}^{2}$ at the maximal ideal at $Q$ and the ideal $(x,y^{2})$ at $P,$ where $x,y$ are local coordinates at $P.$ Now it can be seen that $W$ is non-singular except one point $P_{0}$ over $P$ local ring at $P_{0}$ is isomorphic to localisation of ${\mathbb{C}}[x,y,t]/{(xt+y^{2})}$ at the maximal ideal $(x,y,t).$ Thus $W$ has $A_{1}$ type singularity at $P_{0}.$

If $V$ is as in ii) then the scheme $W$ is isomorphic to blow-up of ${\mathbb{P}}^{2}$ along the ideal $(x,y^{3})$ at $P,$ where $x,y$ are the local coordinates at $P.$ Hence $W$ is non-singular except one point $P_{0}$ over $P$ local ring at $P_{0}$ is isomorphic to localisation of ${\mathbb{C}}[x,y,t]/{(xt+y^{3})}$ at the maximal ideal $(x,y,t).$ Thus $W$ has $A_{2}$ type singularity at $P_{0}.$ $\hfill{\Box}$

Proof of Theorem 1.1: Note that a hyperplane section of $\mathbb{P}(T_{\mathbb{P}^{2}})$ in its natural embedding in ${\mathbb{P}}^{7}$ is a two dimensional scheme of degree six and is the image of the zero scheme of a non zero section of ${\mathcal{O}}_{\mathbb{P}(T_{\mathbb{P}^{2}})}(1).$ Let $s$ be a non zero section of $T_{\mathbb{P}^{2}}$ and $\phi(s)$ be the corresponding section ${\mathcal{O}}_{\mathbb{P}(T_{\mathbb{P}^{2}})}(1).$

First assume that the zeros of the section $s$ are of the form given by a), b) or c). Then the hypothesis of Lemma 2.5 holds with $V_{1}(s)=\emptyset.$ Therefore the scheme $W$ of zeros of the section $\phi(s)$ is equal to the blow-up $\mathbb{P}^{2}$ along $V=V_{0}(s)$ as in Lemma 2.5 and hence irreducible. If $V_{0}(s)$ as in a) then the $W$ is non-singular and is isomorphic to blow-up of ${\mathbb{P}}^{2}$ along three distinct points. By Lemma 2.7, if $V$ as in b) then $W$ is rational normal surface with exactly one $A_{1}$ type singularity and if $V$ as in c) then $W$ is rational normal surface with exactly one $A_{2}$ type singularity.

From this we conclude that, if the zeros of the section $s$ are of the form given by $a),b)$ or $c)$ in Theorem 2.2, then the image of the zero scheme of $\phi(s)$ is an irreducible surface of degree $6$ in $\mathbb{P}^{6}$ described in the Theorem.

Next assume that the zeros of the section $s$ are of the form given by $d)$ or $e)$ in Theorem 2.2. Then the hypothesis of Lemma 2.5 holds with $V_{1}(s)=\ell$ a line in $\mathbb{P}^{2}$ and $V_{0}(s)$ is a single point. Since

[TABLE]

from Lemma 2.5 we see that the scheme of zeros of the section $\phi(s)$ is equal to $W_{0}\cup W_{1},$ where $W_{0}$ is the blow-up $\mathbb{P}^{2}$ along $V_{0}(s)$ and $W_{1}=\mathbb{P}({\mathcal{O}}_{\ell}(1)\oplus{\mathcal{O}}_{\ell}(2)).$ From this we conclude that, if the zeros of the section $s$ are of the form given by $d)$ or $e)$ in Theorem 2.2 then the image of the zero scheme of $\phi(s)$ is a union of two surfaces, namely the images of $W_{0}$ and $W_{1}.$ Since the image of $W_{1}=\mathbb{P}({\mathcal{O}}_{\ell}(1)\oplus{\mathcal{O}}_{\ell}(2))$ is a surface of degree 3, we conclude that the image of $W_{0}$ is also a surface of degree 3, as claimed in ii) of the Theorem. From the Lemma 2.5 it follows that $W_{0}$ is isomorphic to blow-up of ${\mathbb{P}}^{2}$ at a point. Hence both $W_{0}$ and $W_{1}$ are non-singular surfaces. It is now easy to see that if the section $s$ is of the form given by $d)$ (respectively, by $e)$) in Theorem 2.2, then the zeros of $\phi(s)$ consists of union of two non-singular irreducible surfaces, intersecting along a non-singular rational curve (respectively, intersecting along a union of two non-singular rational curves meeting at a point). This completes the proof of the Theorem. $\hfill{\Box}$

Remark 2.8**.**

It is well known that the variety $\mathbb{P}(T_{\mathbb{P}^{2}})$ can be identified with the homogeneous space ${\text{SL}_{3}(\mathbb{C})}/B,$ where ${\text{SL}_{3}}(\mathbb{C})$ is the special linear group of $3\times 3$ matrices with determinant one over the field of complex numbers and $B$ is the Borel subgroup of upper triangular matrices. Thus $\mathbb{P}(T_{\mathbb{P}^{2}})$ is a homogenous variety.

1) The nonsingular hyperplane sections of $\text{SL}_{3}(\mathbb{C})/B$ described in the Theorem 1.1 are the well known Del Pezzo Surfaces of degree 6 in $\mathbb{P}^{6}.$ The Theorem 1.1 describes all the possible degenerations of the Del Pezzo surface of degree 6 inside the homogeneous space $\text{SL}_{3}(\mathbb{C})/B.$

2) The very ample linear system that we considered here is given by the sum of two homologically inequivalent Schubert divisors in ${\text{SL}_{3}}(\mathbb{C}).$ These inequivalent Schubert are known as fundamental Schubert divisors in ${\text{SL}_{3}(\mathbb{C})}/B.$ In the natural embedding these fundamental Schubert divisors are the degree 3 components of a reducible hyperplane section whose components intersect along the union of two rational curves meeting at a single point in the Theorem 1.1. This can be deduced from the description of Bruhat order on the symmetric group $S_{3}$ (see. [BM] Example 1.2.3) and the description of the Schubert divisors (see. [BM] Proposition 1.2.1). By using the Theorem 1.1 we see that the possible deformations of the union of the two fundamental Schubert divisors in ${\text{SL}_{3}(\mathbb{C})}/B$ are

a) union of two irreducible nonsingular surfaces $W_{1}$ and $W_{2}$ intersecting along a irreducible non-singular rational curve and

[TABLE]

b) irreducible rational normal surface with exactly one singularity which is either of the type $A_{1}$ or $A_{3}.$

c) non-singular Del Pezzo Surfaces of degree 6.

**Acknowledgement: ** We would like to thank University of Lille-1 at Lille, France for its hospitality. The second named author would like to thank the University of Artois at Lens, France, for inviting him for an academic visit.