Examples of blown up varieties having projective bundle structures
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Examples of blown up varieties having projective bundle structures NABANITA RAY1,2 1 The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600 113,
India 2 Homi 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. 14C22.
Primary: 14A10; Secondary: 14C20,
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 it happens. Let Z = P˜n be the blow up of a projective space Pn = 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 projective bundle, i.e. Z P(E), where E = OPn−r (1) ⊕ OrPn−r is a locally free sheaf of rank r + 1 on Pn−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 , say X 0 . Now, we blow up P5 along the subvariety X 0 and we get that P˜ 5X 0 is P3 bundle over P2 (see Theorem 3.1). Also, we describe explicitly the rank four vector bundle E over P2 , such that P˜ 5X 0 P(E) (see Theorem 3.2). Take a generic hyperplane H in P5 , such that X 1 = X 0 ∩ H is a non-singular degree three surface in P4 . We get P˜ 4X 1 P(E 1 ), where E 1 is a rank three bundle over P2 which is a quotient of E (see Theorem 3.4). Similarly, take the generic hyperplane H1 in P4 such
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that X 2 = X 1 ∩ H1 is a twisted cubic in P3 . We prove P˜ 3X 2 P(E 2 ), where E 2 is the rank two bundle over P2 which is a quotient of E 1 . Conversely, we prove that if C is a non-linear subvariety of P3 (i.e. C is not a single 3 has a projective bundle structure, then C has to be twisted point or a line in P3 ) and P˜ C cubic.
2. Notations and definitions We denote by Pn the projective space over the field C of complex numbers. Let be a nonsingular sub-variety of Pn and P˜n is denoted as Pn blown up along . Here, π : P˜n → Pn is the canonical blowing up map, and E is the corresponding exceptional divisor. The Picard group of P˜n is generated by π ∗ (OPn (1)) and E . Let X be a smooth pro
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