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Register a new userThe Hilbert schemes of locally Cohen-Macaulay curves in P^3 may after all be connected
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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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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.
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.
We suggest the possibility that the mysterious dark energy component driving the acceleration of the Universe is the leading term, in the de Sitter temperature, of the free energy density of space-time seen as a quantum gravity coherent state of the gravitational field. The corresponding field theory classically has positive pressure, and can be considered as living on the Hubble horizon, or, alternatively, within the non compact part of the Robertson-Walker metric, both manifolds being characterized by the same scale and degrees of freedom. The equation of state is then recovered via the conformal anomaly. No such interpretation seems to be available for negative {Lambda}.
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Hara Charalambous
, Kostas Karagiannis
, Sotiris Karanikolopoulos andn Aristides Kontogeorgis
2021
We consider a Fermat curve $F_n:x^n+y^n+z^n=1$ over an algebraically closed field $k$ of characteristic $pgeq0$ and study the action of the automorphism group $G=left(mathbb{Z}/nmathbb{Z}timesmathbb{Z}/nmathbb{Z}right)rtimes S_3$ on the canonical ring $R=bigoplus H^0(F_n,Omega_{F_n}^{otimes m})$ when $p>3$, $p mid n$ and $n-1$ is not a power of $p$. In particular, we explicitly determine the classes $[H^0(F_n,Omega_{F_n}^{otimes m})]$ in the Grothendieck group $K_0(G,k)$ of finitely generated $k[G]$-modules, describe the respective equivariant Hilbert series $H_{R,G}(t)$ as a rational function, and use our results to write a program in Sage that computes $H_{R,G}(t)$ for an arbitrary Fermat curve.
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