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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 $T$.
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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 $xin Vbackslash C$.
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
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.
We show that an improvement to the best known quantum lower bound for GRAPH-COLLISION problem implies an improvement to the best known lower bound for TRIANGLE problem in the quantum query complexity model. In GRAPH-COLLISION we are given free access to a graph $(V,E)$ and access to a function $f:Vrightarrow {0,1}$ as a black box. We are asked to determine if there exist $(u,v) in E$, such that $f(u)=f(v)=1$. In TRIANGLE we have a black box access to an adjacency matrix of a graph and we have to determine if the graph contains a triangle. For both of these problems the known lower bounds are trivial ($Omega(sqrt{n})$ and $Omega(n)$, respectively) and there is no known matching upper bound.
The Index Erasure problem asks a quantum computer to prepare a uniform superposition over the image of an injective function given by an oracle. We prove a tight $Omega(sqrt{n})$ lower bound on the quantum query complexity of the non-coherent case of the problem, where, in addition to preparing the required superposition, the algorithm is allowed to leave the ancillary memory in an arbitrary function-dependent state. This resolves an open question of Ambainis, Magnin, Roetteler, and Roland (CCC 2011), who gave a tight bound for the coherent case, the case where the ancillary memory must return to its initial state. The proof is based on evaluating certain Krein parameters of a symmetric association scheme defined over partial permutations. The study of this association scheme may be of independent interest.
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