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The Halphen cubics of order two. (English) Zbl 1372.14005
In the paper under review, the authors study the so-called Roulleau-Urzúa arrangement of cubic curves in the complex projective plane, which has an important meaning in the theory of surfaces of general type [X. Roulleau and G. Urzúa, Ann. Math. (2) 182, No. 1, 287–306 (2015; Zbl 1346.14097)]. The main aim behind this paper is to show that the mentioned arrangement can be described very explicitly using a classical theory of linear series and certain facts known for the so-called dual Hesse arrangement of \(9\) lines and \(12\) triple points. The key observation is that cubics appearing in the construction due to Roulleau-Urzúa are in fact Halphen cubics of order \(n/3\) (see Introduction therein for details). The main result of the paper can be formulated as follows.
Main Theorem. For each \(n \in 3\mathbb{N}\), denote by \(H(n)\) the union of \(\frac{4}{3}(n^{2}-3)\) Halphen cubics of order \(n/3\). The singularities of \(H(n)\) are the following: \(12\) points of multiplicity \(n^{2}-3\) at the vertices of the dual Hesse arrangement of lines, \(n^{2}/3 - 1\) triple points infinitely near to them, and \((n^{2}-3)(n^{2}/3 - 3)\) quadruple points.One of possible applications of the Roulleau-Urzúa arrangement of cubics can be noticed in a different context, the so-called bounded negativity conjecture. It can be shown that the Harbourne index of \(H(n)\), the self-intersection of the strict transform of \(H(n)\) under the blowing-up at the singular locus of \(H(n)\) divided by the number of singular points, tends to \(-4\) as \(n \rightarrow \infty\). Also, the authors provide specific equations (for the first time) of the Halphen cubics of order \(2\).
14C20 Divisors, linear systems, invertible sheaves
14E05 Rational and birational maps
14H52 Elliptic curves
14J26 Rational and ruled surfaces
14K12 Subvarieties of abelian varieties
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