Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userWhen the positivity of the h-vector implies the Cohen-Macaulay property
96
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
Let P be a lattice polytope with $h^*$-vector $(1, h^*_1, h^*_2)$. In this note we show that if $h_2^* leq h_1^*$, then $P$ is IDP. More generally, we show the corresponding statements for semi-standard graded Cohen-Macaulay domains over algebraically closed fields.
For a partition $lambda$ of $n$, let $I^{rm Sp}_lambda$ be the ideal of $R=K[x_1, ldots, x_n]$ generated by all Specht polynomials of shape $lambda$. We show that if $R/I^{rm Sp}_lambda$ is Cohen--Macaulay then $lambda$ is of the form either $(a, 1, ldots, 1)$, $(a,b)$, or $(a,a,1)$. We also prove that the converse is true if ${rm char}(K)=0$. To show the latter statement, the radicalness of these ideals and a result of Etingof et al. are crucial. We also remark that $R/I^{rm Sp}_{(n-3,3)}$ is NOT Cohen--Macaulay if and only if ${rm char}(K)=2$.
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 $d geq 2s-1$.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
