Tri-linear birational maps in dimension three

A tri-linear rational map in dimension three is a ϕ : ( P mathvariant="double-struck">C 1 stretchy="false">) 3 stretchy="false">⇢ \phi : (\mathbb {P}_\mathbb {C}^1)^3 \dashrightarrow \mathbb {C}^3 defined by four polynomials without common factor. If alttext="phi"> encoding="application/x-tex">\phi admits an inverse negative 1"> −<!-- − ^{-1}</mml:annotation> , it birational map. In this paper, we address computational and geometric aspects about these transformations. We give characterization of birationality based on the first syzygies entries. More generally, describe all possible minimal graded free resolutions ideal generated With respect to geometry, show that set alttext="German B German i r comma right-parenthesis"> mathvariant="fraktur">B mathvariant="fraktur">i mathvariant="fraktur">r , encoding="application/x-tex">\mathfrak {Bir}_{(1,1,1)} maps, up composition with automorphism alttext="double-struck encoding="application/x-tex">\mathbb locally closed algebraic subset Grassmannian alttext="4"> 4 encoding="application/x-tex">4 -dimensional subspaces vector space polynomials, has eight irreducible components. Additionally, group action given automorphisms alttext="left-parenthesis cubed"> encoding="application/x-tex">(\mathbb {C}^1)^3 defines 19 orbits, each orbits determines isomorphism class base loci