发表机构
Technion–Israel Institute of Technology; Guangdong Technion–Israel Institute of Technology（以色列理工学院; 广东以色列理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在秩四格中引入仿射球面质量并证明其与稳定自交和最小次数的双边不等式，结合高斯-运动学公式与仿射规范化原理，给出四维横截性估计及协变性，并探讨最优常数问题。
AI 中文摘要
设 $F$ 是秩四格中的一个有效平衡有理二维扇。我们引入仿射球面质量 \\[ M_*(F)=\inf_{covol_g(N)=1}M_g(F) \\] 并证明双边比较 \\[ \sqrt{\frac83}\\,m(F)\leq M_*(F)\leq 3\sqrt{2\\,q(F)}. \\] 这里 $q(F)=deg(F\cdot F)$ 是稳定自交，$m(F)$ 是 $F$ 在原始秩二格商下的最小次数。对于满射 $A$，有 $deg(A_*F)=deg(F\cdot[\ker A_R])$；因此 $m(F)$ 等价于与一个完备权一有理二维平面的最小稳定交次数。特别地，\\[ q(F)\geq\frac4{27}m(F)^2. \\] 证明结合了混合稳定交的精确高斯-运动学公式与平稳球面链的仿射规范化原理。在四维中，恒等映射 $Gr^+(2,4)\simeq S^2\times S^2$ 将环境各向同性转化为尖锐的二阶矩横截性估计。我们证明 $M_*$ 的仿射协变性，推导横截性估计的等号与稳定性条件，并给出环境各向同性与零混合交共存的例子，以及六维空间中具有零自交的三扇。确定最优数值常数仍待解决。
英文摘要
Let $F$ be an effective balanced rational two-dimensional fan in a rank-four lattice. We introduce an affine spherical mass \[ M_*(F)=\inf_{covol_g(N)=1}M_g(F) \] and prove the two-sided comparison \[ \sqrt{\frac83}\,m(F)\leq M_*(F)\leq 3\sqrt{2\,q(F)}. \] Here $q(F)=deg(F\cdot F)$ is stable self-intersection and $m(F)$ is the least degree of $F$ under a primitive rank-two lattice quotient. For a surjection $A$, one has $deg(A_*F)=deg(F\cdot[\ker A_R])$; hence $m(F)$ is equivalently the least stable intersection degree with a complete weight-one rational two-plane. In particular, \[ q(F)\geq\frac4{27}m(F)^2. \] The proof combines an exact Gaussian--kinematic formula for mixed stable intersection with an affine normalization principle for stationary spherical links. In dimension four, the identification $Gr^+(2,4)\simeq S^2\times S^2$ converts ambient isotropy into a sharp second-moment transversality estimate. We prove affine covariance for $M_*$, derive equality and stability conditions for the transversality estimate, and give examples where ambient isotropy coexists with zero mixed intersection, as well as a three-fan in six-space with zero self-intersection. Determining the optimal numerical constant remains open.
