In the paper under review, the author studies the so-called cylinders in rational surfaces. Let \(S\) be a smooth rational surface, then a cylinder in \(S\) is an open subset \(U \subset S\) such that \(U \cong \mathbb{C}^{1} \times Z\) for an affine curve \(Z\). It seems to be very difficult to describe all cylinders in a given surface \(X\), so one can consider a similar problem for polarized surfaces. Fix an ample \(\mathbb{Q}\)-divisor \(A\) on the surface \(S\). An \(A\)-polar cylinder in \(S\) is a Zariski open subset \(U\) in \(S\) such that
(A) \(U \cong \mathbb{C}^{1} \times Z\) for some affine curve \(Z\), i.e., \(U\) is a cylinder in \(S\);
(B) There is an effective \(\mathbb{Q}\)-divisor \(D\) on \(S\) such that \(D \sim_{\mathbb{Q}} A\) and \(U = S \setminus \mathrm{Supp}(D)\). Define \[\mu_{A} :=\{ \lambda \in \mathbb{Q}_{>0} \, : \, \text{ the } \,\, \mathbb{Q}\text{-divisor}\,\, K_{S} + \lambda \cdot A \,\, \text{ is } \, \, \text{pseudoeffective}\}.\] Let \(\triangle_{A}\) be the smallest extremal face of the Mori cone \(\overline{\mathrm{NE}(S)}\) that contains \(K_{S} + \mu_{A} \cdot A\). Denote the dimension of the face \(\triangle_{A}\) by \(r_{A}\). Observe that \(r_{A} = 0\) if and only if \(S\) is a smooth del Pezzo surface and \(\mu_{A} \cdot A \sim_{\mathbb{Q}} -K_{S}\). The number \(r_{A}\) is known as the Fujita rank of the divisor \(A\).
During a conference in 2016, Ciliberto asked the following interesting question.
Question. Let \(S\) be a rational surface that is obtained from \(\mathbb{P}^{2}\) by blowing up points in general position, and let \(A\) be an ample \(\mathbb{Q}\)-divisor on \(S\) such that \(r_{A} + K_{S}^{2} \leq 3\). Is it true that \(S\) does not contain \(A\)-polar cylinders?
In the present paper, the author shows a nice result as below.
Main Theorem. Let \(S\) be a smooth rational surface that satisfies the following generality condition:
\((*)\) the self-intersection of every smooth rational curve in \(S\) is at least \(-1\).
Let \(A\) be an ample \(\mathbb{Q}\)-divisor on \(S\) and let \(r_{A}\) be the Fujita rank of the divisor \(A\). Suppose that \(r_{A} + K_{S}^{2} \leq 3\). Then \(S\) does not contain \(A\)-polar cylinders.
Corollary. Let \(S\) be a smooth rational surface that satisfies \((*)\), \(A\) an ample \(\mathbb{Z}\)-divisor on \(S\), and let \(r_{A}\) be the Fujita rank of the divisor \(A\). Define \[V =\mathrm{Spec}\bigg( \bigoplus_{n\geq 0} H^{0}(S, \mathcal{O}_{S}(nA))\bigg).\] Suppose that \(r_{A} + K_{S}^{2} \leq 3\). Then \(V\) does not admit an effective action of the additive group \(\mathbb{C}_{+}\).