Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Numerical Macaulification   An unpublished example due to Joe Harris from 1983 (or earlier) gave two
smooth space curves with the same Hilbert function, but one of the curves was
arithmetically Cohen-Macaulay (ACM) and the other was not. Starting with an
arbitrary homogeneous ideal in any number of variables, we give two
constructions, each of which produces, in a finite number of steps, an ideal
with the Hilbert function of a codimension two ACM subscheme. We call the
subscheme associated to such an ideal "numerically ACM." We study the
connections between these two constructions, and in particular show that they
produce ideals with the same Hilbert function. We call the resulting ideal from
either construction a "numerical Macaulification" of the original ideal.
Specializing to the case where the ideals are unmixed of codimension two, we
show that (a) every even liaison class, $\mathcal L$, contains numerically ACM
subschemes, (b) the subset, $\mathcal M$, of numerically ACM subschemes in
$\mathcal L$ has, by itself, a Lazarsfeld-Rao structure, and (c) the numerical
Macaulification of a minimal element of $\mathcal L$ is a minimal element of
$\mathcal M$. Finally, if we further restrict to curves in $\mathbb P^3$, we
show that the even liaison class of curves with Hartshorne-Rao module
concentrated in one degree and having dimension $n$ contains smooth,
numerically ACM curves, for all $n \geq 1$. The first (and smallest) such
example is that of Harris. A consequence of our results is that the knowledge
of the Hilbert function of an integral curve alone is not enough to decide
whether it contains zero-dimensional arithmetically Gorenstein subschemes of
arbitrarily large degree.

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The categories of this paper are: gr-qc
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The categories of this paper are: econ
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The categories of this paper are: plasm-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: cs
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The categories of this paper are: nucl-th
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The categories of this paper are: eess
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The categories of this paper are: ao-sci
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The categories of this paper are: astro-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: stat
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The categories of this paper are: acc-phys
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The categories of this paper are: alg-geom
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-ph