Hirzebruch-Type Inequalities Viewed as Tools in Combinatorics

The main purpose of this survey is to provide an introduction, algebro-topological in nature, to Hirzebuch-type inequalities for plane curve arrangements in the complex projective plane. These inequalities gain more and more interest due to their utility in many combinatorial problems related to point or line arrangements in the plane. We would like to present a summary of the technicalities and also some recent applications, for instance in the context of the Weak Dirac Conjecture. We also advertise some open problems and questions.

A Note on the Weak Dirac Conjecture

We show that every set $\mathcal{P}$ of $n$ non-collinear points in the plane contains a point incident to at least $\lceil\frac{n}{3}\rceil+1$ of the lines determined by $\mathcal{P}$.

A Pseudoline Counterexample to the Strong Dirac Conjecture

We demonstrate an infinite family of pseudoline arrangements, in which an arrangement of $n$ pseudolines has no member incident to more than $4n/9$ points of intersection. This shows the "Strong Dirac" conjecture to be false for pseudolines.We also raise a number of open problems relating to possible differences between the structure of incidences between points and lines versus the structure of incidences between points and pseudolines.

A New Approach Treating Constrained Systems Based on the Consistency Condition of Constraints

A new approach treating constrained systems based on the consistency condition of constraints is proposed in this paper. This method is simpler and more rigorous in mathematics. It is not necessary to introduce the concepts of weakly equal, strongly equal, Dirac brackets and Dirac conjecture. The procedure of our method is demonstrated by using Cawley's counterexample to Dirac's conjecture.

FADDEEV-JACKIW APPROACH TO GAUGE THEORIES AND INEFFECTIVE CONSTRAINTS

The general conditions for the applicability of the Faddeev–Jackiw approach to gauge theories are studied. When the constraints are effective, a new proof in the Lagrangian framework of the equivalence between this method and the Dirac approach is given. We find, however, that the two methods may give different descriptions for the reduced phase space when ineffective constraints are present. In some cases the Faddeev–Jackiw approach may lose some constraints or some equations of motion. We believe that this inequivalence can be related to the failure of the Dirac conjecture (that says that the Dirac Hamiltonian can be enlarged to an Extended Hamiltonian including all first class constraints, without changes in the dynamics) and we suggest that when the Dirac conjecture fails, the Faddeev–Jackiw approach fails to give the correct dynamics. Finally we present some examples that illustrate this inequivalence.