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Cylinders in singular del Pezzo surfaces. (English) Zbl 1360.14020

In the paper under review, the authors study whether del Pezzo surfaces \(S\) with du Val singularities admit a \((-K_{S})\) -polar cylinders (or equivalently whether they admit a non-trivial \(\mathbb{G}_{a}\)-action on their generalised cones).
Let us recall that for a projective variety \(X\) and an ample divisor \(H\) the generalised cone over the polarised variety \((X,H)\) is the affine variety defined by \[ \hat{X} = \text{Spec} \bigg( \bigoplus_{n\geq 0}H^{0}(X, \mathcal{O}_{X}(nH))\bigg). \]
Also, we need to define the so-called \(M\)-polar cylinders. If \(M\) is a \(\mathbb{Q}\)-divisor on a projective normal variety \(X\), an \(M\)-polar cylinder in \(X\) is an open subset \[ U = X \setminus \text{Supp}(D) \] defined by an effective \(\mathbb{Q}\)-divisor \(D\) in the \(\mathbb{Q}\)-linear equivalence class of \(M\) such that \(U\) is isomorphic to \(Z \times \mathbb{A}^{1}\) for some affine variety \(Z\).
As it was proved by T. Kishimoto et al. [Algebr. Geom. 1, No. 1, 46–56 (2014; Zbl 1418.14016)], the existence of a non-trivial \(\mathbb{G}_{a}\) on the generalised cone over \((X,H)\) is equivalent to the existence of an \(H\)-polar cylinder on \(X\), and thus the main result of the paper can be summed up by the following.
Main Theorem. Let \(S_{d}\) be a del Pezzo surface of degree \(d\) with at most du Val singularities.
1) The surface \(S_{d}\) does not admit a \((-K_{S_{d}})\)-polar cylinder when:
a) \(d=1\) and \(S_{d}\) allows only singular points of types \(A_{1}, A_{2}, A_{3}, D_{4}\) if any;
b) \(d=2\) and \(S_{d}\) allows only singular points of type \(A_{1}\) if any;
c) \(d=3\) and \(S_{d}\) allows no singular point.
2) The surface \(S_{d}\) has a \((-K_{S_{d}})\)-polar cylinder if it is not one of the Del Pezzo surfaces listed in 1).
Corollary. Let \(S_{d}\) be a del Pezzo surface of degree \(d\) with at most du Val singularities. Then the affine cone over \((S_{d},-K_{S_{d}})\) does not admit a non-trivial \(\mathbb{G}_{a}\)-action exactly when:
1) \(d=1\) and \(S_{d}\) allows only singular points of types \(A_{1}, A_{2}, A_{3}, D_{4}\) if any;
2) \(d=2\) and \(S_{d}\) allows only singular points of type \(A_{1}\) if any;
3) \(d=3\) and \(S_{d}\) allows no singular point.

MSC:

14C20 Divisors, linear systems, invertible sheaves
14E05 Rational and birational maps
14J17 Singularities of surfaces or higher-dimensional varieties
14J26 Rational and ruled surfaces
14J45 Fano varieties
14R20 Group actions on affine varieties
14R25 Affine fibrations

Citations:

Zbl 1418.14016
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References:

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