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- W2798282510 abstract "It remains an open problem to classify the Hilbert functions of double points in $mathbb{P}^2$. Given a valid Hilbert function $H$ of a zero-dimensional scheme in $mathbb{P}^2$, we show how to construct a set of fat points $Z subseteq mathbb{P}^2$ of double and reduced points such that $H_Z$, the Hilbert function of $Z$, is the same as $H$. In other words, we show that any valid Hilbert function $H$ of a zero-dimensional scheme is the Hilbert function of a set of a positive number of double points and some reduced points. For some families of valid Hilbert functions, we are also able to show that $H$ is the Hilbert function of only double points. In addition, we give necessary and sufficient conditions for the Hilbert function of a scheme of a double points, or double points plus one additional reduced point, to be the Hilbert function of points with support on a star configuration of lines." @default.
- W2798282510 created "2018-05-07" @default.
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- W2798282510 date "2018-04-26" @default.
- W2798282510 modified "2023-09-27" @default.
- W2798282510 title "Hilbert functions of schemes of double and reduced points" @default.
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