Related papers: Counting point configurations in projective space
This paper begins by extending the notion of a combinatorial configuration of points and lines to a combinatorial configuration of points and planes that we refer to as configurations of order $2$. We then proceed to investigate a further…
Symplectic invariants introduced in math-ph/0702045 can be computed for an arbitrary spectral curve. For some examples of spectral curves, those invariants can solve loop equations of matrix integrals, and many problems of enumerative…
Cross-ratio degrees count configurations of points $z_1,\ldots, z_n \in \mathbb{P}^1$ satisfying $n - 3$ cross-ratio constraints, up to isomorphism. These numbers arise in multiple contexts in algebraic and tropical geometry, and may be…
One way to characterize configurations of points up to congruence is by considering the distribution of all mutual distances between points. This paper deals with the question if point configurations are uniquely determined by this…
We introduce the notion of order projections using the order unit property of a positive element in an order unit space and characterize them in terms of (geometric) orthogonality. We describe order projections of the order unit space…
We give a new proof of a result of DiPippo and Wan for counting points of bounded height on projective spaces over global function fields. The new proof adapts the geometry of numbers arguments used by Schanuel in the number field case.
We study bipartite maps on the plane with one infinite face and one face of perimeter 2. At first we consider the problem of their enumeration an then study the connection between the combinatorial structure of a map and the degree of its…
We study combinatorial configurations with the associated point and line graphs being strongly regular. Examples not belonging to known classes such as partial geometries and their generalizations or elliptic semiplanes are constructed.…
The classical theory of the cross-ratio is a beautiful case study of the moduli of ordered points of the projective line and of invariants of the action of $PGL_2$. We generalize the theory of the cross-ratio to the setting of $S$-valued…
Let $(C,p_1,\ldots,p_n)$ be a fixed general pointed curve and let $(X,x_1,\ldots,x_n)$ be a smooth hypersurface of degree $e$ and dimension $r$ with $n$ general points. We consider the problem of enumerating maps $f:C\to X$ of degree $d$…
We identify certain Gromov-Witten invariants counting rational curves with given incidence and tangency conditions with the Betti numbers of moduli spaces of point configurations in projective spaces. On the Gromov-Witten side, S. Fomin and…
We consider the configuration space of planar $n$-gons with fixed perimeter, which is diffeomorphic to the complex projective space $\mathbb{C}P^{n-2}$. The oriented area function has the minimal number of critical points on the…
We consider the problem of counting the number of possible sets of rankings (called ranking patterns) generated by unfolding models of codimension one. We express the ranking patterns as slices of the braid arrangement and show that all…
We explain connections among several, a priori unrelated, areas of mathematics: combinatorics, algebraic statistics, topology and enumerative algebraic geometry. Our focus is on discrete invariants, strongly related to the theory of…
We consider the set of points in projective $n$-space that generate an extension of degree $e$ over given number field $k$, and deduce an asymptotic formula for the number of such points of absolute height at most $X$, as $X$ tends to…
Optimization under structural constraints is typically analyzed through projection or penalty methods, obscuring the geometric mechanism by which constraints shape admissible dynamics. We propose an operator-theoretic formulation in which…
Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…
This paper deals with projective shape analysis, which is a study of finite configurations of points modulo projective transformations. The topic has various applications in machine vision. We introduce a convenient projective shape space,…
We introduce two notions of a contractive orbit of a set-valued map defined in a first countable space. The first defines the contraction with respect to the topology of the underlying space while the second defines the contraction with…
We observe that some natural mathematical definitions are lifting properties relative to simplest counterexamples, namely the definitions of surjectivity and injectivity of maps, as well as of being connected, separation axioms $T_0$ and…