Negative curves of small genus on surfaces (Q2182413)

In this very interesting paper, the authors study negative curves in algebraic surfaces. Let \(X\) be a smooth geometrically irreducible projective surface over a field \(\Bbbk\). A negative curve in \(X\) is a complete, reduced, irreducible curve with negative self-intersection. Let us recall that if \(X\) is a smooth complex projective surface then the bounded negativity conjecture predicts that there exists an integer \(b(X)\) such that for every irreducible and reduced curve \(C \subset X\) one has \(C^{2} \geq -b(X)\). In the paper under review the authors study a slightly different question. Question. Let \(X\) be a smooth geometrically irreducible projective surface over an arbitrary field \(\Bbbk\). For which integers \(g\geq 0\) are there infinitely many negative curves in \(X\) of genus \(g\)? In order to formulate the main result of the paper we need to recall that \(b_{1}(X)= \dim_{\mathbb{Q}_{p}}H^{1}_{\mathrm{et}}(X,\mathbb{Q}_{p})\) for any prime \(p\) different from \(\mathrm{char}(\Bbbk)\), and the Picard number \(\rho(X)\) is the rank over \(\mathbb{Z}\) of the Néron-Severi group \(\mathrm{NS}(X)\) of \(X\). Main Result. Let \(X\) be a smooth geometrically irreducible projective surface over a field \(\Bbbk\). There are effective constants \(c_{1}\) and \(c_{2}\) for which the number \(\tau(X)\) of negative curves \(C \subset X\) with geometric genus \(g(C) < b_{1}(X)/4\) is finite and bounded above by \(c_{1}e^{c_{2}\rho(X)}\). One has \(\tau(X) \leq 2^{0.902 \rho(X)}\) for sufficiently large \(\rho(X)\). It is worth emphasizing that the above result is sharp in positive characteristic due to the Frobenius morphism. In order to prove the above result the authors develop the theory of hyperbolic codes -- using such techniques is somehow surprising and this idea can be considered as a very valuable guidepost for further developments.