Suppose $f,g$ are homogeneous polynomials of degree $d$ defining smooth hypersurfaces $X_f = V(f)\subset \mathbb{P}^{m-1}$ and $X_g = V(g)\subset\mathbb{P}^{n-1}$. Then the sum $f(x)+g(y)$ defines a smooth hypersurface $X=V(f(x)+g(y))\subset\mathbb{P}^{m+n-1}$ with an action of $\mu_d$ scaling the $g$ variables. Motivated by the work of Orlov, we construct a semi-orthogonal decomposition of the derived category of coherent sheaves on $[X/\mu_d]$ provided $d\geq \mathrm{max}\{m,n\}$.

中文翻译：

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与两个势能之和相关联的等变派生类别

假设 $f,g$ 是定义光滑超曲面 $X_f = V(f)\subset \mathbb{P}^{m-1}$ 和 $X_g = V(g)\subset\mathbb 的阶次为 $d$ 的齐次多项式{P}^{n-1}$。然后总和 $f(x)+g(y)$ 定义了一个平滑的超曲面 $X=V(f(x)+g(y))\subset\mathbb{P}^{m+n-1}$ 与$\mu_d$ 缩放 $g$ 变量的操作。受 Orlov 工作的启发，我们在 $[X/\mu_d]$ 上构建了相干滑轮的派生类别的半正交分解，提供 $d\geq \mathrm{max}\{m,n\}$。