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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 $xin Vbackslash C$.
A complex surface $S$ is said to be isogenous to a product if $S$ is a quotient $S=(C_1 times C_2)/G$ where the $C_i$s are curves of genus at least two, and $G$ is a finite group acting freely on $C_1 times C_2$. In this paper we classify all regular
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
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
Let $S$ be a surface isogenous to a product of curves of unmixed type. After presenting several results useful to study the cohomology of $S$ we prove a structure theorem for the cohomology of regular surfaces isogenous to a product of unmixed type w
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