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Register a new userOn the arithmetically Cohen-Macaulay property for sets of points in multiprojective spaces
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We study the arithmetically Cohen-Macaulay (ACM) property for finite sets of points in multiprojective spaces, especially $(mathbb P^1)^n$. A combinatorial characterization, the $(star)$-property, is known in $mathbb P^1 times mathbb P^1$. We propose a combinatorial property, $(star_n)$, that directly generalizes the $(star)$-property to $(mathbb P^1)^n$ for larger $n$. We show that $X$ is ACM if and only if it satisfies the $(star_n)$-property. The main tool for several of our results is an extension to the multiprojective setting of certain liaison methods in projective space.
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We give examples of infinitely extendable (not as cones) arithmetically Cohen-Macaulay and arithmetically Gorenstein subvarieties of projective spaces and which are not complete intersections. The proof uses the computation of the dimension of the Hilbert scheme of codimension $2$ subschemes of projective spaces due to G. Ellingsrud and of arithmetically Gorenstein codimension $3$ subschemes due to J. O. Kleppe and R.-M. Mir{o}-Roig.
In this article, we provide a complete list of simple Cohen-Macaulay codimension 2 singularities together with a list of adjacencies which is complete in the case of fat point and space curve singularities.
Let $P_{text{MAX}}(d,s)$ denote the maximum arithmetic genus of a locally Cohen-Macaulay curve of degree $d$ in $mathbb{P}^3$ that is not contained in a surface of degree $<s$. A bound $P(d, s)$ for $P_{text{MAX}}(d,s)$ has been proven by the first author in characteristic zero and then generalized in any characteristic by the third author. In this paper, we construct a large family $mathcal{C}$ of primitive multiple lines and we conjecture that the generic element of $mathcal{C}$ has good cohomological properties. With the aid of emph{Macaulay2} we checked the validity of the conjecture for $s leq 100$. From the conjecture it would follow that $P(d,s)= P_{text{MAX}}(d,s)$ for $d=s$ and for every $d geq 2s-1$.
We study relations between the Cohen-Macaulay property and the positivity of $h$-vectors, showing that these two conditions are equivalent for those locally Cohen-Macaulay equidimensional closed projective subschemes $X$, which are close to a complete intersection $Y$ (of the same codimension) in terms of the difference between the degrees. More precisely, let $Xsubset mathbb P^n_K$ ($ngeq 4$) be contained in $Y$, either of codimension two with $deg(Y)-deg(X)leq 5$ or of codimension $geq 3$ with $deg(Y)-deg(X)leq 3$. Over a field $K$ of characteristic 0, we prove that $X$ is arithmetically Cohen-Macaulay if and only if its $h$-vector is positive, improving results of a previous work. We show that this equivalence holds also for space curves $C$ with $deg(Y)-deg(C)leq 5$ in every characteristic $ch(K) eq 2$. Moreover, we find other classes of subschemes for which the positivity of the $h$-vector implies the Cohen-Macaulay property and provide several examples.
Progress on the problem whether the Hilbert schemes of locally Cohen-Macaulay curves in projective 3 space are connected has been hampered by the lack of an answer to a question that was raised by Robin Hartshorne in his paper On the connectedness of the Hilbert scheme of curves in projective 3 space Comm. Algebra 28 (2000) and more recently in the open problems list of the 2010 AIM workshop Components of Hilbert Schemes available at http://aimpl.org/hilbertschemes: does there exist a flat irreducible family of curves whose general member is a union of d disjoint lines on a smooth quadric surface and whose special member is a locally Cohen-Macaulay curve in a double plane? In this paper we give a positive answer to this question: for every d, we construct a family with the required properties, whose special fiber is an extremal curve in the sense of Martin-Deschamps and Perrin. From this we conclude that every effective divisor in a smooth quadric surface is in the connected component of its Hilbert scheme that contains extremal curves.
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