2021/04/29 by Olivia Dumitrescu, Dumitrescu, Olivia, Rick Miranda +1
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2104.14141
openalex publication_date 2021/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study (i)-curves with i∈ \-1, 0, 1\ in the blown up projective space ℙr 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 counterexample to Hilbert's 14th problem. We introduce the notion of classes of (0)- and (1)-curves in ℙr 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, (that 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 ℙr with r+3 points blown up. As an application, we use the technique of movable curves to reprove that if F2≤ 0 then Y is not a Mori Dream Space and we propose to apply this technique to other spaces.