Construction of birational trilinear volumes via tensor rank criteria

Abstract

We discuss algebraic, geometric, and constructive aspects of birational maps φ : (P 1 ) 3 99K P 3 defined by trilinear polynomials f0, f1, f2, f3. In particular, we establish a novel connection between birationality and tensor rank. First, we analyze the syzygies of the defining polynomials of φ. Specifically, we provide the list of all the possible minimal Z 3 -graded free resolutions of the base ideal I = (f0, f1, f2, f3) ⊂ R, where R = C[s0, s1] ⊗ C[t0, t1] ⊗ C[u0, u1]. Secondly, we describe the space of these transformations. Namely, it is an algebraic subset of Gr(4, R(1,1,1)) × P 15 C with eight irreducible components. Additionally, we provide the complete list of the isomorphism classes of the possible base loci of φ. Our strategy relies on the factorization of φ as the composition of a canonical rational map ζ : (P 1 C ) 3 −→ P n C , for some n > 3, and a linear projection π : P n C −→ P 3 C . We explain how the group action given by Aut(P 1 C ) 3 on (P 1 C ) 3 extends to the image of ζ. This new group action determines finitely many orbits, which correspond to the isomorphism classes. Regarding the construction, we define φ by means of control points and weights. For adequately constrained control points, we prove that birationality is achieved if and only if a certain 2 × 2 × 2 tensor has rank one. Interestingly, birationality boils down to a simpler rank condition suitable for numerical computations.   

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