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Register a new userA spectral lower bound for the divisorial gonality of metric graphs
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Let $Gamma$ be a compact metric graph, and denote by $Delta$ the Laplace operator on $Gamma$ with the first non-trivial eigenvalue $lambda_1$. We prove the following Yang-Li-Yau type inequality on divisorial gonality $gamma_{div}$ of $Gamma$. There is a universal constant $C$ such that [gamma_{div}(Gamma) geq C frac{mu(Gamma) . ell_{min}^{mathrm{geo}}(Gamma). lambda_1(Gamma)}{d_{max}},] where the volume $mu(Gamma)$ is the total length of the edges in $Gamma$, $ell_{min}^{mathrm{geo}}$ is the minimum length of all the geodesic paths between points of $Gamma$ of valence different from two, and $d_{max}$ is the largest valence of points of $Gamma$. Along the way, we also establish discre
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According to a well-know theorem by Sturm, a vibrating string is divided into exactly N nodal intervals by zeros of its N-th eigenfunction. Courant showed that one half of Sturms theorem for the strings applies to the theory of membranes: N-th eigenfunction cannot have more than N domains. He also gave an example of a eigenfunction high in the spectrum with a minimal number of nodal domains, thus excluding the existence of a non-trivial lower bound. An analogue of Sturms result for discretizations of the interval was discussed by Gantmacher and Krein. The discretization of an interval is a graph of a simple form, a chain-graph. But what can be said about more complicated graphs? It has been known since the early 90s that the nodal count for a generic eigenfunction of the Schrodinger operator on quantum trees (where each edge is identified with an interval of the real line and some matching conditions are enforced on the vertices) is exact too: zeros of the N-th eigenfunction divide the tree into exactly N subtrees. We discuss two extensions of this result in two directions. One deals with the same continuous Schrodinger operator but on general graphs (i.e. non-trees) and another deals with discrete Schrodinger operator on combinatorial graphs (both trees and non-trees). The result that we derive applies to both types of graphs: the number of nodal domains of the N-th eigenfunction is bounded below by N-L, where L is the number of links that distinguish the graph from a tree (defined as the dimension of the cycle space or the rank of the fundamental group of the graph). We also show that if it the genericity condition is dropped, the nodal count can fall arbitrarily far below the number of the corresponding eigenfunction.
Let $(S,mathcal L)$ be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle $mathcal L$ of degree $d > 35$. In this paper we prove that $K^2_Sgeq -d(d-6)$. The bound is sharp, and $K^2_S=-d(d-6)$ if and only if $d$ is even, the linear system $|H^0(S,mathcal L)|$ embeds $S$ in a smooth rational normal scroll $Tsubset mathbb P^5$ of dimension $3$, and here, as a divisor, $S$ is linearly equivalent to $frac{d}{2}Q$, where $Q$ is a quadric on $T$.
Let $(S,mathcal L)$ be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle $mathcal L$ of degree $d > 25$. In this paper we prove that $chi (mathcal O_S)geq -frac{1}{8}d(d-6)$. The bound is sharp, and $chi (mathcal O_S)=-frac{1}{8}d(d-6)$ if and only if $d$ is even, the linear system $|H^0(S,mathcal L)|$ embeds $S$ in a smooth rational normal scroll $Tsubset mathbb P^5$ of dimension $3$, and here, as a divisor, $S$ is linearly equivalent to $frac{d}{2}Q$, where $Q$ is a quadric on $T$. Moreover, this is equivalent to the fact that the general hyperplane section $Hin |H^0(S,mathcal L)|$ of $S$ is the projection of a curve $C$ contained in the Veronese surface $Vsubseteq mathbb P^5$, from a point $xin Vbackslash C$.
By considering graphs as discrete analogues of Riemann surfaces, Baker and Norine (Adv. Math. 2007) developed a concept of linear systems of divisors for graphs. Building on this idea, a concept of gonality for graphs has been defined and has generated much recent interest. We show that there are connected graphs of treewidth 2 of arbitrarily high gonality. We also show that there exist pairs of connected graphs ${G,H}$ such that $Hsubseteq G$ and $H$ has strictly lower gonality than $G$. These results resolve three open problems posed in a recent survey by Norine (Surveys in Combinatorics 2015).
Here we prove that the Hilbert-Kunz mulitiplicity of a quadric hypersurface of dimension $d$ and odd characteristic $pgeq 2d-4$ is bounded below by $1+m_d$, where $m_d$ is the $d^{th}$ coefficient in the expansion of $mbox{sec}+mbox{tan}$. This proves a part of the long standing conjecture of Watanabe-Yoshida. We also give an upper bound on the HK multiplicity of such a hypersurface. We approach the question using the HK density function and the classification of ACM bundles on the smooth quadrics via matrix factorizations.
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