A lower bound for $K^2_S$

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 $K^2_S\geq -d(d-6)$. 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 $T\subset \mathbb P^5$ 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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