On ( i )-Curves in Blowups of P r

In this paper, we study ( i ) -curves with i ∈ { − 1 , 0 , 1 } in the blown-up projective space P r in general points. The notion of ( − 1 ) -curves was analyzed in the early days of mirror symmetry by Kontsevich, with the motivation of counting curves on a Calabi–Yau threefold. In dimension two, Nagata studied planar ( − 1 ) -curves in order to construct a counterexample to Hilbert’s 14th problem. We introduce the notion of classes of ( 0 ) - and ( 1 ) -curves in P r with s points blown up, and we prove that their number is finite if and only if the space is a Mori Dream Space. We further introduce a bilinear form on a space of curves and a unique symmetric Weyl-invariant class, F (which we will refer to as the anticanonical curve class ). For Mori Dream Spaces, we prove that ( − 1 ) -curves can be defined arithmetically by the linear and quadratic invariants determined by the bilinear form. Moreover, ( 0 ) - and ( 1 ) -Weyl lines give the extremal rays for the cone of movable curves in P r with r + 3 points blown up. As an application, we use the technique of movable curves to reprove that if F 2 ≤ 0 then Y is not a Mori Dream Space, and we propose to apply this technique to other spaces.