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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 $$2$$.
##### MSC:
 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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