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Examples of Blown up Varieties Having Projective Bundle Structures

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Examples of Blown up Varieties Having Projective Bundle Structures 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.
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