论文标题
维度三
Tri-linear birational maps in dimension three
论文作者
论文摘要
维度第三的三线性理性图是一个有理图$ ϕ：（\ Mathbb {p} _ \ Mathbb {C}^1）^3 \ dashrightarrow \ dashrightArrow \ dashrightArrow \ Mathbb {p} _ \ Mathbb {C}^3 $由四个tri-linear polynomials定义。如果$ ϕ $承认一个反理性地图$ ϕ^{ - 1} $，则是一张三线性的birational地图。在本文中，我们解决了有关这些转换的计算和几何方面。我们基于条目的第一个共融来给出异性恋的特征。更普遍地，我们描述了这些条目产生的理想的所有可能最小的自由分辨率。关于几何形状，我们表明了$ \ mathfrak {bir} _ {（（1,1,1,1）} $的tri linear birational图，直至组成，其自动形态为$ \ mathbb {p} _ \ mathbb {c}^c}^c}^3 $，是$ sprosp的$ s $ suble subseim n $ suble nemenn in $三线性多项式的矢量空间，并具有八个不可还原组件。 Additionally, the group action on $\mathfrak{Bir}_{(1,1,1)}$ given by composition with automorphisms of $(\mathbb{P}_\mathbb{C}^1)^3$ defines 19 orbits, and each of these orbits determines an isomorphism class of the base loci of these transformations.
A tri-linear rational map in dimension three is a rational map $ϕ: (\mathbb{P}_\mathbb{C}^1)^3 \dashrightarrow \mathbb{P}_\mathbb{C}^3$ defined by four tri-linear polynomials without a common factor. If $ϕ$ admits an inverse rational map $ϕ^{-1}$, it is a tri-linear birational map. In this paper, we address computational and geometric aspects about these transformations. We give a characterization of birationality based on the first syzygies of the entries. More generally, we describe all the possible minimal graded free resolutions of the ideal generated by these entries. With respect to geometry, we show that the set $\mathfrak{Bir}_{(1,1,1)}$ of tri-linear birational maps, up to composition with an automorphism of $\mathbb{P}_\mathbb{C}^3$, is a locally closed algebraic subset of the Grassmannian of $4$-dimensional subspaces in the vector space of tri-linear polynomials, and has eight irreducible components. Additionally, the group action on $\mathfrak{Bir}_{(1,1,1)}$ given by composition with automorphisms of $(\mathbb{P}_\mathbb{C}^1)^3$ defines 19 orbits, and each of these orbits determines an isomorphism class of the base loci of these transformations.
