DESSINS D'ENFANTS AND HYPERSURFACES WITH MANY $A_j$-SINGULARITIES

Abstract

Abstract. We show the existence of surfaces of degree d in È 3 ( ) with approximately 3j+2 6j(j+1) d 3 singularities of type A j , 2 ≤ j ≤ d − 1. The result is based on Chmutov's construction of nodal surfaces. For the proof we use plane trees related to the theory of Dessins d'Enfants.Our examples improve the previously known lower bounds for the maximum number µ A j (d) of A j -singularities on a surface of degree d in most cases. We also give a generalization to higher dimensions which leads to new lower bounds even in the case of nodal hypersurfaces in È n , n ≥ 5.To conclude, we work out in detail a classical idea of B. Segre which leads to some interesting examples, e.g. to a sextic with 36 cusps. IntroductionAll possible configurations of singularities on a surface of degree 3 in È 3 := È 3 ( ) are known since Schläfli's work [25] in the 19 th century, see [14] and [18] for explicit equations and illustrating pictures. In the case of degree 4, the classification was completed recently by Yang [29] using computers.Much less is known for higher degrees, even when restricting to a particular type of singularity. E.g., the maximum number of A 1 -singularities on a surface of degree d is only known for d ≤ 6. We recently improved the case d = 7 using computer algebra and geometry over prime fields [19]. The best lower bounds for surfaces of large degree d with A 1 -singularities are given by Chmutov's construction [9].For higher singularities -e.g., singularities of type A j which are locally equivalent to x j+1 + y 2 + z 2 -the situation is even more difficult. We denote by µ Aj (d) the maximum number of singularities of type A j a surface of degree d in È 3 can have. Barth [5] constructed a quintic with 15 singularities of type A 2 (also called (ordinary) cusps), and the author constructed a sextic with 35 such singularities [20] using computer algebra in characteristic zero which showed µ A2 (6) ≥ 35. The detailed study of a generalization of an idea of B. Segre [26] which we give in the appendix leads to: µ A2 (6) ≥ 36 (see equations (11) and (12), and corollary 12). Recently, Barth and others considered the codes connected to surfaces with threedivisible cusps in analogy to the codes related to even sets of nodes, see [4,6,23].In general, the best lower bounds for the µ Aj

Real Line Arrangements and Surfaces with Many Real Nodes

“…give hypersurfaces in È n , n ≥ 5, which improve the previously known lower bounds for the maximum number of nodes in higher dimensions slightly (see [16]; [3] gives a detailed discussion of all these folding polynomials and their critical points).…”

Disjoinable Lagrangian spheres and dilations

“…There are numerous constructions of hypersurfaces with many nodes in the literature, see for instance [9, vol. 2, p. 419] or [35,Section 8.1]. On the other hand, the size of collections obtained in this way is limited by known upper bounds for the possible number of Date: September 1, 2013. nodes, such as those derived in [58] using Hodge-theoretic methods.…”

On the Solution of the Multi-Asset Black-Scholes Model: Correlations, Eigenvalues and Geometry

“…In fact, exists a whole surface inside the hypercube, where the determinant of ρ vanishes. This surface, is called the Kummer surface Σ K in algebraic geometry [19], [20], [21], [22], is defined by the equation…”

“…In this paper, we study the multi-asset Black-Scholes model in terms of the importance that the correlation parameter space (which is equivalent to an N dimensional hypercube) has in the solution of the option pricing problem. We show that inside of this hypercube there is a surface, called the Kummer surface Σ K [19], [20], [21], [22], where the determinant of the correlation matrix ρ is zero, so over Σ K the usual formula for the propagator of the N asset Black-Scholes equation is no longer valid. Worse than that, outside this surface, the are points where the determinant of ρ becomes negative, so the usual propagator becomes complex and divergent.…”