= Euler defect identity for a projective line arrangement
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{title2=$t_2=3+\sum_{k\ge4}(k-3)t_k+\sum_{j\ge4}(j-3)f_j$}
Let $t_k$ count vertices where $k$ lines meet and $f_j$ count faces with $j$ sides in a nonpencil projective line arrangement. Using <Euler characteristic> one, $v=\sum t_k$, $e=\sum kt_k$, and $2e=\sum jf_j$ gives $t_2=3+\sum_{k\ge4}(k-3)t_k+\sum_{j\ge4}(j-3)f_j$. Thus few double vertices control both high-multiplicity vertices and nontriangular faces. Discarding the face term gives the usual arrangement form of the ordinary-line inequality.
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