Purdy's Inequality for Hyperplane Arrangements
Prior state unknown→disproved
the refined form for essential arrangements in projective three-space
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geometry-topology / Discrete geometry
Purdy's Inequality for Hyperplane Arrangements
For an arrangement of $n$ hyperplanes in $\mathbb{P}^3_{\mathbb{C}}$ with $\ell$ intersection lines and $p$ intersection points where at least three hyperplanes meet, the refined form of Purdy's inequality expects $p - \ell + n + 2 \ge 0$. An explicit arrangement built from roots of unity violates it.
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For an arrangement of $n$ hyperplanes in $\mathbb{P}^3_{\mathbb{C}}$ with $\ell$ intersection lines and $p$ intersection points where at least three hyperplanes meet, the refined form of Purdy's inequality expects $p - \ell + n + 2 \ge 0$. An explicit arrangement built from roots of unity violates it.
the refined form for essential arrangements in projective three-space
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