Examples of Blown up Varieties Having Projective Bundle Structures

Home , Blowing up, Projective bundle, Twisted cubic

Proc. Indian Acad. Sci. (Math. Sci.) (2020) 130:15 https://doi.org/10.1007/s12044-019-0532-6

Examples of blown up varieties having projective bundle structures

NABANITA RAY1,2

1The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600 113, India 2Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400 094, India Email: [email protected]

MS received 1 May 2019; revised 29 May 2019; accepted 30 May 2019

Abstract. We give some examples of blow up of projective space along some projective subvariety, such that these blown up spaces are isomorphic to a projective bundle over some projective space.

Keywords. Blow up; projective bundle; nef cone; chow ring.

2000 Mathematics Subject Classiﬁcation. Primary: 14A10; Secondary: 14C20, 14C22.

1. Introduction It is always interesting to ask, under which criterion, blow up of a projective variety along a projective subvariety is isomorphic to the projective bundle over some projective variety. In general, blow up of a projective space along a projective subvariety is not isomorphic to the projective bundle over some projective space. But we know some examples, where ˜ n n it happens. Let Z = P be the blow up of a projective space P = PV along a linear subspace Pr−1. It is well known from Section 9.3.2 of [2] that Z is a total space of P(E) E = O − ( ) ⊕ Or projective bundle, i.e. Z , where Pn r 1 Pn−r is a locally free sheaf of rank r + 1onPn−r . Motivated by this result, we produce here some non-linear examples, where blow up of a projective space along some non-linear subvariety will be isomorphic to a projective bundle over a projective variety. Also, we have calculated the nef cone of those varieties. We take the three-fold P1 ×P2 in P5 by Segre embedding. This is degree three three-fold in P5,sayX .Now,weblowupP5 along the subvariety X and we get that P˜ 5 is P3 0 0 X0 bundle over P2 (see Theorem 3.1). Also, we describe explicitly the rank four vector bundle E over P2, such that P˜ 5 P(E) (see Theorem 3.2). X0 5 Take a generic hyperplane H in P , such that X1 = X0 ∩ H is a non-singular degree three surface in P4. We get P˜ 4 P(E ), where E is a rank three bundle over P2 which X1 1 1 4 is a quotient of E (see Theorem 3.4). Similarly, take the generic hyperplane H1 in P such

© Indian Academy of Sciences 0123456789().: V,-vol 15 Page 2 of 11 Proc. Indian Acad. Sci. (Math. Sci.) (2020) 130:15 that X = X ∩ H is a twisted cubic in P3. We prove P˜ 3 P(E ), where E is the rank 2 1 1 X2 2 2 2 two bundle over P which is a quotient of E1. Conversely, we prove that if C is a non-linear subvariety of P3 (i.e. C is not a single P3 P˜ 3 point or a line in ) and C has a projective bundle structure, then C has to be twisted cubic.

2. Notations and definitions We denote by Pn the projective space over the field C of complex numbers. Let be a non- n ˜ n n ˜ n n singular sub-variety of P and P is denoted as P blown up along . Here, π : P → P is the canonical blowing up map, and E is the corresponding exceptional divisor. The ˜ n ∗ Picard group of P is generated by π (OPn (1)) and E. Let X be a smooth projective variety, E a holomorphic vector bundle on it, and by P(E) the projectivization of E, defined as P(E) = Proj(Sym(E)), where Sym(E) is the symmetric algebra of the sheaf of the section of E.Also,P(E) can be described as a projective bundle of the one-dimensional quotient of E. We follow here the definition of [4] to describe P(E).

3. Main results 1 2 5 Let i : P × P → P be the Segre embedding defined by sending [x0, x1]×[y0, y1, y2] to [x0 y0, x0 y1, x0 y2, x1 y0, x1 y1, x1 y2].Let{zi | i = 0, 1,...,5} be the homogeneous 5 1 2 5 co-ordinate of P .Ifi(P × P ) = X0, then X0 is a three-fold in P which is defined by the equations f0 = z0z4 − z1z3, f1 = z0z5 − z2z3, f2 = z2z4 − z1z5. The morphism i is defined by the complete linear system (1, 1) of P1 ×P2, where the Picard group of P1 ×P2 5 is Z ⊕ Z. Hence deg(X0) = (1, 1) · (1, 1) · (1, 1) = 3inP .

5 3 2 Theorem 3.1. P blown up along the closed subscheme X0 is a P bundle over P .Here the P3 bundle map φ˜ : P˜ 5 → P2 is defined by the divisor 2π ∗ H − E . 0 X0 X0 |O ( )⊗I | Proof. Let us consider the linear system P5 2 X0 , which is defined by set of all degree P5 I two hypersurfaces of , which contain X0. Here, X0 is the ideal sheaf corresponding to 0(O ( ) ⊗ I ) the closed subscheme X0. The vector space H P5 2 X0 is generated by the basis, , , |O ( )⊗I | P2 f0 f1 f2. Hence the linear system P5 2 X0 is isomorphic to . So the linear system defines a morphism φ : P5\X → P2 and we can extend this map as φ˜ : P˜ 5 → P2 0 0 0 X0 such that the following diagram commutes:

P˜ 5 π P5 X0 φ 0 φ0 P2

and the φ˜ map is defined by the linear system |2π ∗ H − E | of P˜ 5 . 0 X0 X0 ˜ 3 Now, our claim is that each fiber of φ0 is isomorphic to P . First, define the φ0 map coordinate-wise, i.e. φ0([z0, z1,...,z5]) =[z0z4 − z1z3, z0z5 − z2z3, z2z4 − z1z5]= [ , , ] φ−1[ , , ]= ( , ) ⊆ ( , ) ( ( , )) = f0 f1 f2 .Now 0 1 0 0 V f1 f2 and X0 V f1 f2 .deg V f1 f2 4. As Proc. Indian Acad. Sci. (Math. Sci.) (2020) 130:15 Page 3 of 11 15

( ) = ( , ) = ∪ = P3 φ˜−1[ , , ] deg X0 3, V f1 f2 X0 L such that deg L 1, i.e. L . Hence 0 1 0 0 is = φ˜−1[ , , ] isomorphic to the strict transformation of L. Similarly, when a0 0, then 0 a0 a1 a2 is a strict transformation of V (a0 f1 − a1 f0, a0 f2 − a2 f0)\X0. When a1 = 0, then φ˜−1[ , , ] ( − , − )\ 0 a0 a1 a2 is a strict transformation of V a1 f0 a0 f1 a2 f1 a1 f2 X0 and when = φ˜−1[ , , ] ( − , − )\ a2 0, 0 a0 a1 a2 is a strict transformation of V a0 f2 a2 f0 a1 f2 a2 f1 X0. ˜ Now, we need to check if the φ0 map satisfies the co-cycle condition over an affine cover P2 P2 =∪2 ={ = } = of ([4], Chapter II.7, Exercise 10). Let i=0Ui , where Ui ai 0 .Leta0 0, 3 3 V (a0 f1−a1 f0, a0 f2−a2 f0) = X0∪P , where P = V (g1, g2), and gi are the hyperplanes 5 in P . Then, a0 f1 − a1 f0 = h11g1 + h12g2 and a0 f2 − a2 f0 = h21g1 + h22g2, where { } ψ : φ˜−1( ) → hij are degree one and linearly independent set. Define the morphism 0 0 U0 × P3 ∈ φ˜−1( ) φ˜ ( ) =[ , , ] U0 .Letx 0 U0 , where 0 x a0 a1 a2 and x corresponds to the point 3 b in P over [a0, a1, a2]. Then we define the map ψ0(x) =[a0, a1, a2]×b. Now define λ : ×P3 → φ˜−1( ) [ , , ]×[ , , , ]∈ ×P3 0 U0 0 U0 .Let a0 a1 a2 b0 b1 b2 b3 U0 . Then we can find g1 and g2 such that V (g1, g2) ⊂ V (a0 f1 − a1 f0, a0 f2 − a2 f0), where fi and gi are the same 3 ˜ as defined above. Here [b0, b1, b2, b3]∈P V (g1, g2) = L L. Then [b0, b1, b2, b3] φ˜−1( ) [ , , ] ψ ◦ λ = corresponds to a point in 0 U0 on the fiber of a0 a1 a2 . Hence, clearly 0 0 id and λ0 ◦ ψ0 = id. Similarly, ψ1 and ψ2 are defined. =[ , , ]∈ ∩ ψ ◦ ψ−1 : Now, consider a a0 a1 a2 U0 U1. We need to show the map 1 0 a × P3 → a × P3 is a linear automorphism. But, this will clearly follow from the fact that, if a0 f1 − a1 f0 = h11g1 + h12g2 and a0 f2 − a2 f0 = h21g1 + h22g2, then a2 f1 − a1 f2 = a2 (a f − a f ) + a1 (a f − a f ) = ( a2 h + a1 h )g + ( a2 h + a1 h )g . Hence a0 0 1 1 0 a0 0 2 2 0 a0 11 a0 21 1 a0 12 a0 22 2 the result is proved.

Now, P˜ 5 = P(E), where E is a rank four vector bundle over P2. Then φ˜∗ H = 2π ∗ H − X0 0 P2 E X0 , where H is the hyperplane section of . Picard group as well as Neron Severi group P( ) φ˜∗ O ( ) π ∗( ) = O ( )⊗φ˜∗( ) of E is generated by 0 H and P(E) 1 .So H P(E) n1 0 n2 H . Note that = π ∗( )3( π ∗( )− )2 = (O ( )+φ˜∗( ))3φ˜∗( )2. = 1 H 2 H E X0 P(E) n1 0 n2 H 0 H Hence n1 1 which π ∗( ) = O ( ) ⊗ φ˜∗( ) = ⊗ P( ) P( ) implies H P(E) 1 0 n2 H .IfwetakeE E n2 H , then E E ˜∗ ˜ 5 and OP(E)(n1) = OP(E)(n1) ⊗ φ (n2 H ). So w.l.o.g., we can consider P = P(E) such ∗ 0 X0 that π (H) = OP(E)(n), n > 0. As π is a generically ﬁnite map, deg(π) = 1 implies that ∗ 5 5 5 ∗ (π (H)) = 1 ⇒ n · OP(E)(1) = 1 ⇒ n = 1, i.e. π (H) ∼ OP(E)(1).

Theorem 3.2. If P(E) is as above, then E is the cokernel of an injective homomorphism O (− ) ⊕ O (− ) −→ O6 P2 1 P2 1 P2 , i.e. we have an exact sequence

−→ O (− ) ⊕ O (− ) −→ O6 −→ −→ . 0 P2 1 P2 1 P2 E 0 (1)

∗ Proof. As π (H) ∼ OP(E)(1),themapπ is given by the line bundle OP(E)(1) and then 0 0 2 0 2 we have 6 = dim H (P(E), OP(E)(1)) = dim H (P ,π∗(OP(E)(1))) = dim H (P , E). 2 As OP(E)(1) is globally generated, E is also a globally generated vector bundle of P . O6 F Hence, we get a surjective morphism from P2 to E, where is the kernel, i.e.,

−→ F −→ O6 −→ −→ . 0 P2 E 0 (2) 15 Page 4 of 11 Proc. Indian Acad. Sci. (Math. Sci.) (2020) 130:15

So, clearly H 0(P2, F) = 0 and H 1(P2, F) = 0. Now, consider another exact sequence of vector bundle of P2:

−→ G −→ 2(O6 ) −→ 2( ) −→ . 0 S P2 S E 0 (3)

Here S2(F) is the second symmetric power of any locally free sheaf F. Corresponding 2(O6 ) to the above exact sequence (2), we can get a ﬁltration of S P2 ([4], Chapter II.5, Exercise 16),

2(O6 ) = M ⊇ M ⊇ M ⊇ M = S P2 0 1 2 3 0(4)

2( ) M /M 2(O6 )/M M G F ⊗ M /M such that S E 0 1 S P2 1, then 1 .Also, E 1 2, 2 S (F) M2. Using the above isomorphism, we have

0 −→ S2(F) −→ G −→ F ⊗ E −→ 0. (5)

Tensor the exact sequence (2) by the locally free sheaf F,

−→ F ⊗ F −→ O6 ⊗ F −→ ⊗ F −→ . 0 P2 E 0 (6)

We have the surjective map F ⊗ F → S2(F) with kernel ∧2(F), the second exterior power F ⊗ F → ⊗ F O6 ⊗ F → G of and the identity map E E . This induces a map P2 such that the following diagram commutes:

−→ F ⊗ F −→ O6 ⊗ F −→ ⊗ F −→ 0 ⏐ P2⏐ E ⏐ 0 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ (7) 0 −→ S2(F) −→ G −→ E ⊗ F −→ 0.

O6 ⊗ F → G ∧2(F) By the construction, the middle column map P2 is surjective and is contained in the kernel. As the rank of the kernel is one, the kernel is exactly ∧2(F). Hence, we have the exact sequence

−→ ∧ 2(F) −→ O6 ⊗ F → G −→ . 0 P2 0 (8)

φ˜∗ = π ∗ − = π ∗ − φ˜∗ ∼ We have 0 H 2 H E X0 , which implies that E X0 2 H 0 H O ( ) ⊗ φ˜∗O (− ) P(E) 2 0 P2 1 . The dimension of the global section of an exceptional divisor is 0 ˜ 5 0 ˜∗ one. Hence, 1 = dim H (P , E ) = dim H (P(E), OP( )(2)⊗φ OP2 (−1)).Usingthe X0 X0 E 0 0 2 2 projection formulas (3) and (8)fromabove,wehave1 = dim H (P , S (E)⊗OP2 (−1)) = dim H 1(P2, G(−1)) = dim H 2(P2, ∧2(F)(−1)). Hence, the only possibility of ∧2(F) is 2 OP2 (−2). We know any rank two bundle of P can be written as

0 −→ OP2 (m) −→ F −→ OP2 (n) ⊗ IZ −→ 0, (9)

2 where Z is the dimension zero closed subscheme of P and IZ is the corresponding ideal sheaf ([3], Chapter 2). Proc. Indian Acad. Sci. (Math. Sci.) (2020) 130:15 Page 5 of 11 15

Here in our case,

−2H = c1(F) = (m + n)H , c2(F) = mn + l(Z). (10)

4 We claim that l(Z) = 0 and m = n =−1. From (2), ∧ (E) = OP2 (2), i.e. c1(E) = 2H . Let ζ = OP(E)(1). Hence, we have the equation

4 3 2 ζ = c1(E)ζ − c2(E)ζ . (11)

We know ζ 5 = 1, then

4 3 1 = c1(E)ζ − c2(E)ζ . (12)

Substituting ζ 4 in (12), we get

2 3 1 = (4H − c2(E))ζ ⇒ c2(E) = 3. (13)

−1 2 −1 Now, from (2), we have c(E)·c(F) = c(OP2 ) ⇒ c(F) = c(E) = (1+2H +3H ) = 2 1 − 2H + H ⇒ c2(F) = 1. So from (10), it is clear that either n = m =−1, then we are done or m < −1; n ≥ 0. Now, assume m < −1 and n ≥ 0. Tensoring (10)byOP(1),wehave

0 −→ OP2 (m + 1) −→ F(1) −→ OP2 (n + 1) ⊗ IZ −→ 0. (14)

Here c1(F(1)) = 0 and c2(F(1)) = 0. Then we can apply Theorem 4.14.(iv) of [3] which 2 states that E is a rank two bundle of P with c1(E) = 0. If E is stable, then c2(E) ≥ 2. So, in our case, F(1) is not stable. Then from Theorem 4.14.(i) of [3], we have m + 1 ≥ 0, hence m ≥−1. This contradicts our assumption. Finally, we get F = OP2 (−1) ⊕ OP2 (−1) as there are no non-linear extension of the vector bundle F by the line bundle over P2. Hence the result follows.

Remark 3.3. We know from Theorem 9.6 of [2] that the Chow ring of P(E) is

A(P2)[ζ ] A(P(E)) = , 4 ∗ 3 ∗ 2 (15) ζ + c1(E )ζ + c2(E )ζ

ζ ∼ π ∗( ) ∼ O ( ) (P2) = Z[α] α ∼ where H P(E) 1 . Also, we have A α3 , where H .Usingthe following short exact sequence,

−→ ∗ −→ O6 −→ O ( ) ⊕ O ( ) = −→ 0 E P2 P2 1 P2 1 F 0 (16) and the Whitney sum formula, we get

( ∗) = (O6 ). ( )−1 = 1 = − + 2 2. ct E ct 2 ct F 1 2H t 3H t (17) P (1 + H t)2 15 Page 6 of 11 Proc. Indian Acad. Sci. (Math. Sci.) (2020) 130:15

Z[α,ζ ] Hence, the Chow ring of P˜ 5 or P(E) is , ζ,α ∈ A1(P˜ 5 ) and E ∼ X0 α3,ζ 4−2αζ3+3α2ζ 2 X0 X0 2ζ − α.

The bundle OP2 (1) ⊕ OP2 (1) has six independent sections and it can be generated by four sections. Let us take ﬁve independent global sections of OP2 (1) ⊕ OP2 (1), which O5 → O ( ) ⊕ O ( ) generate this rank two bundle. Then we get a surjection P2 P2 1 P2 1 , where ∗ E1 is the kernel of the map. Hence we have the exact sequence

−→ O (− ) ⊕ O (− ) −→ O5 −→ −→ . 0 P2 1 P2 1 P2 E1 0 (18)

Theorem 3.4. Let E be as above. Then P(E ) P˜ 4 , where X is the hyperplane section 1 1 X1 1 of X in P5, i.e. X is a cubic surface in P4.Ifπ : P˜ 4 → P4 is the blown up map and 0 1 1 X1 φ˜ : P( ) → P2 φ˜∗(O ( )) ∼ π ∗O ( ) − 1 E1 is the projectivization map, then 1 P2 1 2 1 P4 1 E X1 .

Proof. We have the following commutative diagram:

−→ ∗ −→ O5 −→ O ( ) ⊕ O ( ) −→ 0 E⏐ ⏐P2 P2 1 ⏐ P2 1 0 ⏐1 ⏐ ⏐ ⏐ ⏐ ⏐ (19) −→ ∗ −→ O6 −→ O ( ) ⊕ O ( ) −→ . 0 E P2 P2 1 P2 1 0

As the right and the middle arrows are identity and injective respectively, the left arrow also exists and is injective, which implies that E → E1 is surjective. This corresponds to : P( )→ P( ) O ( ) ∗O ( ) the inclusion i E1 E such that P(E1) 1 i1 P(E) 1 . ˜ 2 Let φ1 : P(E1) → P be the projection morphism. From (18), it is clear that E1 is 0 2 4 globally generated and h (P , E1) = 5. Then, we have a morphism π1 : P(E1) → P by O ( ) the line bundle P(E1) 1 , such that the following diagram commutes:

i1 P(E1) P(E)

π1 π P4 i P5

∗ Here i is the canonical inclusion map, i (OP5 (1)) = OP4 (1). In Proposition 3.1,wehave 5 2 deﬁned a rational map φ0 : P P , given by φ0([z0, z1,...,z5]) =[z0z4−z1z3, z0z5− 4 5 z2z3, z2z4 − z1z5]. Now, consider P as a hyperplane of P , given by the equation z0 = z4. 4 2 Then we get a rational map φ1 : P P , which is φ0([z0, z1, z2, z3, z5]) =[z0z4 − , 2 − , − ] φ P4 z1z3 z0 z2z3 z2z4 z1z0 .Themap 1 is not deﬁned on the cubic surface X1 of = − = 2 − = − I given by the equations g0 z0z4 z1z3, g1 z0 z2z3, g2 z2z4 z1z0.Let X1 be the P4 π −1I · O P( ) ideal sheaf of X1 in . 1 X1 P(E1) is an invertible sheaf of ideal on E1 , because π −1I · O = −1(π −1I · O ) π −1I · O 1 X1 P(E1) i1 X0 P(E) and X0 P(E) is an invertible sheaf of ideal on P(E) = P˜ 5 . By the Universal Property of Blowing Up ([4], Chapter II.7), we have X0 the unique morphism P(E ) → P˜ 4 over P4. 1 X1 Proc. Indian Acad. Sci. (Math. Sci.) (2020) 130:15 Page 7 of 11 15

Now, φ can be extended as φ˜ : P˜ 4 → P2 and deﬁned by the linear system 1 1 X1 π ∗O ( ) − π : P˜ 4 → P4 2 1 P4 1 E X1 . We have the natural blowing up map 1 X . Here, clearly ∗ ∗ 1 φ˜ → φ˜ → π ∗(O ( )) 1 E 1 E1 1 P4 1 is a surjective map. This corresponds to a morphism P˜ 4 → P(E ) over P2 ([4], Chapter II.7). Hence we have the commutative diagram X1 1

˜ 4 P(⏐E1) −→ P⏐ −→ P(⏐E1) ⏐ ⏐X1 ⏐ ⏐ ⏐ ⏐ (20) P(E) −→ P˜ 5 −→ P(E). X0

The composition of the lower horizontal arrows is identity and the vertical maps are inclusion. Then P(E ) → P˜ 4 → P(E ) is also an identity. Similarly, P˜ 4 → P(E ) → 1 X1 1 X1 1 P˜ 4 is also an identity. Hence we have P˜ 4 P(E ). X1 X1 1

˜ Z[α,ζ ] Remark 3.5.A(P4 ) = A(P(E )) = , where ζ ∼ OP( )(1) and α ∼ X1 1 α3,ζ 3−2αζ2+3α2ζ E1 ∗ φ˜ ( )ζ,α∈ 1(P( )) ∼ ζ − α 1 H A E1 and E X1 2 .

COROLLARY 3.6

If we have the following short exact sequence:

−→ ∗ −→ O4 −→ O ( ) ⊕ O ( ) −→ , 0 E2 P2 P2 1 P2 1 0 (21) then P˜ 3 P(E ), where X is a twisted cubic in P3, which can be obtained by cutting X2 2 2 down the cubic surface in P4 by a hyperplane.

Proof. The proof will follow similarly as we have done in Theorem 3.4.

Remark 3.7. The Chow ring of P3 blown up along the twisted cubic is A(P˜ 3 ) = X2 Z[α,ζ ] ∗ ˜ ∗ A(P(E )) = , where ζ ∼ OP( )(1) ∼ π (O 3 (1)) and α ∼ φ (H ) 2 α3,ζ 2−2αζ+3α2 E2 2 P 2 ζ,α ∈ 1(P( )) ∼ ζ − α φ˜ : P( ) → P2 A E2 and E X2 2 . Here, 2 E2 is a natural projective bundle map and π : P˜ 3 → P3 is the blowing up map. 2 X2

Theorem 3.8. Let C be an irreducible subvariety of P3 other than the linear subspaces P3 P˜ 3 = ( , , ) of . C has a projective bundle structure if C V f0 f1 f2 , where fi are irreducible 2 3 homogeneous polynomials,deg( fi ) = deg( f j ) = d and d1 = deg(C) = d − 1 in P .

P˜ 3 P˜ 3 P( ) → Pn π Proof. Let C has projective bundle structure, i.e. C E , where is the projectivization map and n ≤ 2. Let C = V ({gi | i = 0, 1,...,r}).Ifdeg(gi ) = deg(g j ) for i = j, then we can = ({ | = , ,..., }) = ni construct fi such that C V fi i 0 1 r and fi gi for some positive integer ni such that the degree of each fi is the same. 15 Page 8 of 11 Proc. Indian Acad. Sci. (Math. Sci.) (2020) 130:15

Consider the following diagram:

P˜ 3 I P( ) C E φ π P3 Pn

Here I is the isomorphism and φ is the blow up map. ψ : P3 Pn is the corresponding 3 rational map which is deﬁned on P \C, i.e. ψ is given by the linear system |OP3 (d) ⊗ IC|, where d = deg( fi )>1 and IC is the ideal sheaf corresponding to the subvariety C. π 1 −1 Case I (n=1). P(E) −→ P , rank(E) = 3 and C = V ( f0, f1). Clearly, ψ ([a0, a1]) = −1 −1 V (a0 f1 − a1 f0) and then φ (ψ ([a0, a1])) = V (a0 f1 − a1 f0)\C ∪ EC , where V (a0 f1 − a1 f0)\C is the strict transformation of V (a0 f1 − a1 f0)\C and EC is the exceptional divisor corresponding to the blow-up map φ. This gives π −1(a) = 3 V (a0 f1 − a1 f0)\C Sd which is degree d hypersurface in P which is not isomorphic to P2. Hence we get a contradiction. This implies C will never be a complete intersection curve in P3. π 2 Case II (n=2). P(E) −→ P , rank(E) = 2 and C = V ( f0, f1, f2). Clearly, −1 ψ ([a0, a1, a2]) = V (a0 f1−a1 f0, a0 f2−a2 f0).AsC is an irreducible curve, C becomes −1 −1 an irreducible component of ψ (a), where a =[a0, a1, a2]. Then ψ (a) = C ∪ Ca. −1 −1 Hence φ (ψ (a)) = EC ∪ Ca, where EC is the exceptional divisor corresponding to −1 the blowing up map φ and Ca is the strict transformation of Ca. This gives π (a) = Ca 1 3 and this will be isomorphic to P only when deg(Ca) = 1inP . Hence we get that each fi 2 is a reduced polynomial (otherwise Ca will become a non-reduced curve for some a ∈ P ) 2 2 and deg(C) = d − 1 (as deg(V (a0 f1 − a1 f0, a0 f2 − a2 f0)) = d . Hence the theorem is proved.

PROPOSITION 3.9

P˜ 3 Considering the same notations of Theorem 3.8, C has a projective bundle structure only when C is a genus zero curve.

P˜ 3 P( ) P2 i ( ) Proof. Let C E , where E is a rank 2 vector bundle over .LetA X be the rational equivalence class of codimension i cycle of the scheme X. Here,

1 ∗ A (P(E)) = ZOP(E)(1) ⊕ Zπ OP2 (1) Z ⊕ Z, 2 ∗ ∗ ∗ A (P(E)) = ZOP(E)(1) · π OP2 (1) ⊕ Z(π OP2 (1) · π OP2 (1)) Z ⊕ Z.

Now consider the following commutative diagram:

P( ) j P3 N C π π C i P3 Proc. Indian Acad. Sci. (Math. Sci.) (2020) 130:15 Page 9 of 11 15

3 N is a normal bundle of C in P , P(N) = EC , the exceptional divisor corresponding φ ˜ = φ∗O ( ) ∈ 1(P˜ 3 ) to the blowing up map , i and j are inclusions. Let h P3 1 A C and =[ ]∈ 1(P˜ 3 ) e EC A C . 1(P˜ 3 ) = Z ⊕ Z ˜ 2(P˜ 3 ) From Proposition 13.13 of [2], we have A C e h, A C is generated by 2 =− (O ( )) ˜2 ( ) ∈ 1( ) = π ∗( ) 2(P˜ 3 ) Z ⊕ Z e j∗ P(N) 1 , h and j∗ FD for D A C and FD D . A C if and only if any two point on the curve C is rationally equivalent, i.e. Pic(C) = Z. Hence C is a genus zero curve.

P˜ 3 Theorem 3.10. C has a projective bundle structure if and only if C is a twisted cubic in P3.

Proof. Using the same notations as in Proposition 3.9,wehavePic(P˜ 3 )=Ze ⊕ Zh˜, ∗ C Pic(P(E)) = ZOP(E)(1) ⊕ Zπ OP2 (1) Z ⊕ Z. Rename ζE = OP(E)(1) and ∗ ˜ ˜ H = π OP2 (1), H ∼ dh − e and ζE ∼ h in the Picard group. ζ 2 · = ( ) ζ 2 · = ˜2( ˜ − ) We know E H c1 E . From the above equivalence, we get E H h dh e . This implies

d = c1(E), (22)

3 using the intersection products from Proposition 13.13 of [2]. Also, e =−4d1 −2g+2, 3 2 where from Theorem 3.8, the degree of C in P , d1 = d − 1 and from Proposition 3.9, genus of C = g = 0. Then

3 3 2 (dζE − H) = e =−4(d − 1) + 2 ⇒ 3ζ 3 − 2ζ 2 + ζ 2 − 3 =− 2 + d E 3d E H 3d E H H 4d 6 ⇒− 3 + =− 2 + ( ζ 2 = ( ) = ,ζ 2= 3= ) 2d 3d 4d 6 As E H c1 E d E H 1 and H 0 3 ⇒ 2d3 − 4d2 − 3d + 6 = 0 ⇒ d = 2or ± . 2

Hence d = 2. So C is degree three curve in P3 which is either a cubic in P2 or a twisted cubic in P3.AsC is not a complete intersection curve, C is a twisted cubic. = − 2 = − 2 = − = ( , , ) Clearly, f0 Z0 Z2 Z1, f1 Z1 Z3 Z2, f3 Z0 Z3 Z1 Z2 and C V f0 f1 f2 is a twisted cubic in P3.

Remark 3.11. We know that linear subspace is always a degree one complete intersection variety. But in the above three examples, we have seen that none of X0, X1, X2 are complete intersection variety. So if blow up of projective space along a projective variety is isomorphic to a projective bundle, then this projective variety need not be a complete intersection.

Now, it is interesting to know what are the nef cones of P(E), P(E1) and P(E2). 3 1 3 X2 is a twisted cubic in P . X2 is the image of P into P by the map [u,v]→ [u3, u2v,uv2,v3], where u,vare homogeneous co-ordinates of P1. Also, we can consider the map via P1 × P1, i.e. P1 → P1 × P1 → P3, where P1 → P1 × P1 is given by [u,v]→[u2,v2]×[u,v] and P1 ×P1 → P3 is the Segre embedding. Pic(P1 ×P1)=Z×Z, then X ∼ (2, 1) in P1 × P1. We have the blowing up map π : P˜ 3 → P3.TheNeron 2 2 X2 15 Page 10 of 11 Proc. Indian Acad. Sci. (Math. Sci.) (2020) 130:15

1 ˜ 3 ∗ Severi group, N (P ) is generated by H = π OP3 (1) and the exceptional divisor E . X2 2 X2 The numerical equivalence class of one cycle, N (P˜ 3 ) is generated by the pullback of the 1 X2 P3 = π ∗ general line in , l 2 lP3 and an exceptional curve e, as described in [1]. Then

· = , · = , · = , · =− . l H 1 l E X2 0 e H 0 e EX2 1 (23)

The effective cone of curves NE(P˜ 3 ) ⊂ N (P˜ 3 ) is generated by e and C˜ ∼ al − be, X2 1 X2 ˜ 3 where C is a strict transformation of degree a curve in P which intersect X2 in b points / ˜ ∼ − b > with b a maximum. Claim C l 2e. If the claim is not true, then a 2. Let S be 1 1 3 the image of P × P in P containing the twisted cubic, then S ∼ OP3 (2). So the strict transformation of S is S˜ ∼ 2H − E in P˜ 3 . Then C˜ · S˜ = 2a − b < 0, which implies X2 X2 C ⊂ S.LetC ∼ (α, β) in P1 × P1.degC = α + β = a in P˜ 3 and C · X = 2α + β = b. X2 2 b = 2α + β<2(α + β) = 2a < b, hence the contradiction. So our claim is true, i.e. NE(P˜ 3 ) is generated by e and C˜ ∼ l − 2e. For a more general calculation, see [1]. X2

COROLLARY 3.12

˜ 3 ∗ ∗ Nef cone of P is generated by π OP3 (1) and 2π OP3 (1) − E . X2 2 2 X2

Proof. As we know, the nef cone is a dual of the closure of the effective cone of curves, and hence the result will follow from the above discussion.

COROLLARY 3.13

˜ 4 ∗ ∗ Nef cone of P is generated by π OP4 (1) and 2π OP4 (1) − E . X1 1 1 X1

φ˜ ˜ 4 1 2 ∗ Proof. We know P P(E ) −→ P is deﬁned by the linear system 2π OP4 (1)− E ∼ X2 1 1 X1 ˜ ∗ φ1 OP2 (1) which is nef because pullback of ample is nef. Here our claim is that this is a boundary of the nef cone. This is an effective divisor but not big, as the highest power self-intersection is zero. So our claim is proved. π ∗O ( ) ∼ O ( ) Now, another generator is 1 P4 1 P(E1) 1 . We have the short exact sequence in P2:

0 −→ L −→ E1 −→ E2 −→ 0. (24)

As deg(E1) = deg(E2),deg(L) = 0, i.e. L = OP2 . As the extension of the two nef bundle O ( ) is nef, E1 is a nef vector bundle. Also, P(E1) 1 is a nef. This is not ample because the quotient of ample is ample, but E2 is not ample. Hence the result follows.

COROLLARY 3.14

˜ 5 ∗ ∗ Nef cone of P is generated by π OP5 (1) and 2π OP5 (1) − E . X0 X0

Proof. Same as Corollary 3.13. Proc. Indian Acad. Sci. (Math. Sci.) (2020) 130:15 Page 11 of 11 15

Acknowledgements The author would like to thank Prof. D. S. Nagaraj, IMSc, Chennai for suggesting this prob- lem and for valuable suggestions while the author was working on this project. This work is ﬁnancially supported by a fellowship from IMSc, Chennai (HBNI), DAE, Government of India.

References [1] Blanc J and Lamy S, Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links, Proc. Lond. Math. Soc. (3) 105(5) (2012) 1047–1075 [2] Eisenbud D and Harris J, 3264 and All That—A Second Course in Algebraic Geometry (2016) (Cambridge: Cambridge University Press) [3] Friedman R, Algebraic Surface and Holomorphic Vector Bundles (1998) (New York: Springer) [4] Hartshorne R, Algebraic Geometry (1977) (NewYork: Springer-Verlag)

Communicating Editor: Parameswaran Sankaran