Related papers: Hypersurface Arrangements with Generic Hypersurfac…
We calculate the fibre integrals of the hypersurface in a torus in the form of their Mellin transforms. Especially, our method works efficiently for an affine hypersurface defined by a so called simpliciable polynomial. The relations…
Master character of the multidimensional homogeneous Euler equation is discussed. It is shown that under restrictions to the lower dimensions certain subclasses of its solutions provide us with the solutions of various hydrodynamic type…
A simple and efficient algorithm to numerically compute the genus of surfaces of three-dimensional objects using the Euler characteristic formula is presented. The algorithm applies to objects obtained by thresholding a scalar field in a…
If $X \subset \mathbb P^n$ is a reduced subscheme, we say that $X$ admits an unexpected hypersurface of degree $t$ for multiplicity $m$ if the imposition of having multiplicity $m$ at a general point $P$ fails to impose the expected number…
We describe a framework for estimating Hilbert-Samuel multiplicities $e_XY$ for pairs of projective varieties $X \subset Y$ from finite point samples rather than defining equations. The first step involves proving that this multiplicity…
Tools of Topological Data Analysis provide stable summaries encapsulating the shape of the considered data. Persistent homology, the most standard and well studied data summary, suffers a number of limitations; its computations are hard to…
We find sharp upper bounds on the order of the automorphism group of a hypersurface in complex projective space in every dimension and degree. In each case, we prove that the hypersurface realizing the upper bound is unique up to…
Let $\mathcal{A}$ be a central hyperplane arrangement in $\mathbb{C}^{n+1}$ and $H_i,i=1,2,...,d$ be the defining equations of the hyperplanes of $\mathcal{A}$. Let $f=\prod_i H_i$. There is a global Milnor fibration $F\hookrightarrow…
This paper designs an alogrithm to compute the minimal combinations of finite sets in Euclidean spaces, and applys the algorithm of study the moment maps and geometric invariant stability of hypersurfaces. The classical example of cubic…
We study large uniform random maps with one face whose genus grows linearly with the number of edges. They can be seen as a model of discrete hyperbolic geometry. In the past, several of these hyperbolic geometric features have been…
Hyperuniform particle arrangements are characterized by a local number variance that grows more slowly than the volume of the observation window. We generalize this concept to describe particle systems in which particles carry weights:…
It is known that there exist hyperplane arrangements with same underlying matroid that admit non-homotopy equivalent complement manifolds. In this work we show that, in any rank, complex central hyperplane arrangements with up to 7…
We show that the virtual Euler characteristics of the moduli spaces of $s$-pointed algebraic curves of genus $g$ can be determined from a polynomial in $1/\gamma$ where $\gamma$ permits specialization, through $\gamma=1,$ to the complex…
The full moments expansion of the joint probability distribution of an isotropic random field, its gradient and invariants of the Hessian is presented in 2 and 3D. It allows for explicit expression for the Euler characteristic in ND and…
It is shown that the diffeomorphism type of the complement to a real space line arrangement in any dimensional affine ambient space is determined only by the number of lines and the data on multiple points.
Simplicial arrangements are classical objects in discrete geometry. Their classification remains an open problem but there is a list conjectured to be complete at least for rank three. A further important class in the theory of hyperplane…
In this study we introduce a second type of higher order generalised geometric polynomials. This we achieve by examining the generalised stirling numbers $S(n; k;\alpha;\beta;\gamma)$ [Hsu & Shiue,1998] for some negative arguments. We study…
It is well known that the Euler characteristic of the cohomology of a complex algebraic variety coincides with the Euler characteristic of its cohomology with compact support. An old result of G. Laumon asserts that a relative version of…
We give a characterizaton of smooth ample Hypersurfaces in Abelian Varieties and also describe an irreducible connected component of their moduli space: it consists of the Hypersurfaces of a given polarization type, plus the iterated…
While intersections of convex sets are convex, their unions have rather complicated behavior. Some natural contexts where they appear include duality arguments involving boundaries of convex sets and valuations, which have an Euler…