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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 $T$.
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 (mathc
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 i
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 prove
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
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