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Register a new userConnectivity of hyperplane sections of domains
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Authors
Matteo Varbaro
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During the conference held in 2017 in Minneapolis for his 60th birthday, Gennady Lyubeznik proposed the following problem: Find a complete local domain and an element in it having three minimal primes such that the sum of any two of them has height 2 and the sum of the three of them has height 4. In this note this beautiful problem will be discussed, and will be shown that the principle leading to the fact that such a ring cannot exist is false. The specific problem, though, remains open
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The purpose of this paper is twofold. In the first part we concentrate on hyperplane sections of algebraic schemes, and present results for determining when Grobner bases pass to the quotient and when they can be lifted. The main difficulty to overcome is the fact that we deal with non-homogeneous ideals. As a by-product we hint at a promising technique for computing implicitization efficiently. In the second part of the paper we deal with families of algebraic schemes and the Hough transforms, in particular we compute their dimension, and show that in some interesting cases it is zero. Then we concentrate on their hyperplane sections. Some results and examples hint at the possibility of reconstructing external and internal surfaces of human organs from the parallel cross-sections obtained by tomography.
We introduce a new algebra associated with a hyperplane arrangement $mathcal{A}$, called the Solomon-Terao algebra $mbox{ST}(mathcal{A},eta)$, where $eta$ is a homogeneous polynomial. It is shown by Solomon and Terao that $mbox{ST}(mathcal{A},eta)$ is Artinian when $eta$ is generic. This algebra can be considered as a generalization of coinvariant algebras in the setting of hyperplane arrangements. The class of Solomon-Terao algebras contains cohomology rings of regular nilpotent Hessenberg varieties. We show that $mbox{ST}(mathcal{A},eta)$ is a complete intersection if and only if $mathcal{A}$ is free. We also give a factorization formula of the Hilbert polynomials when $mathcal{A}$ is free, and pose several related questions, problems and conjectures.
We use symplectic techniques to obtain partial results on Mahlers conjecture about the product of the volume of a convex body and the volume of its polar. We confirm the conjecture for hyperplane sections or projections of $ell_p$-balls or the Hanner polytopes.
We present algebraic and geometric arguments that give a complete classification of the rational normal scrolls that are hyperplane section of a given rational normal scrolls.
Let $R = mathbb{K}[x_1, ldots, x_n]$ and $I subset R$ be a homogeneous ideal. In this article, we first obtain certain sufficient conditions for the subadditivity of $R/I$. As a consequence, we prove that if $I$ is generated by homogeneous complete intersection, then subadditivity holds for $R/I$. We then study a conjecture of Avramov, Conca and Iyengar on subadditivity, when $I$ is a monomial ideal with $R/I$ Koszul. We identify several classes of edge ideals of graphs $G$ such that the subadditivity holds for $R/I(G)$. We then study the strand connectivity of edge ideals and obtain several classes of graphs whose edge ideals are strand connected. Finally, we compute upper bounds for multigraded Betti numbers of several classes of edge ideals.
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