Generalizations of Joints Problem

We generalize the joints problem to sets of varieties and prove almost sharp bound on the number of joints. As a special case, given a set of $N$ $2$-planes in $\mathbb{R}^6$, the number of points at which three $2$-planes intersect and span $\mathbb{R}^6$ is at most $CN^{3/2+ε}$. We also get almost sharp bound on the number of joints with multiplicities. The main tools are polynomial partitioning and induction on dimension.

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