Related papers: Postulation for 2-superfat points in the plane
A superposition rule is a particular type of map that enables one to express the general solution of certain systems of first-order ordinary differential equations, the so-called Lie systems, out of generic families of particular solutions…
We characterize Hilbert polynomials that give rise to Hilbert schemes with two Borel-fixed points and determine when the associated Hilbert schemes or their irreducible components are smooth. In particular, we show that the Hilbert scheme…
We study how the supporting hyperplanes produced by the projection process can complement the method of alternating projections and its variants for the convex set intersection problem. For the problem of finding the closest point in the…
We show that the generalized SQG equation on the plane is locally well-posed in spaces of low regularity solutions (essentially H\"older continuous with H\"older exponents depending on the equation parameter $\alpha\in(0,\frac 12)$) that…
This paper introduces complexes of linear varieties, called inclics (for INductively Constructible LInear ComplexeS). As examples, we study galaxies (these are constructed starting with a star configuration to which we add general points in…
We give a linear algebraic and a monadic descriptions of the Hilbert scheme of points on the affine space of dimension $n$ which naturally extends Nakajima's representation of the Hilbert scheme of points on the plane. As an application of…
Two notions of riffle shuffling on finite Coxeter groups are given: one using Solomon's descent algebra and another using random walk on chambers of hyperplane arrangements. These coincide for types $A$,$B$,$C$, $H_3$, and rank two groups.…
Plane arrangements are a useful tool for surface and volume modelling. However, their main drawback is poor scalability. We introduce two key novelties that enable the construction of plane arrangements for complex objects and entire…
We investigate the well-posedness of $\alpha$-SQG equations in the half-plane, where $\alpha=0$ and $\alpha=1$ correspond to the 2D Euler and SQG equations respectively. For $0<\alpha \le 1/2$, we prove local well-posedness in certain…
Given a smooth curve on a smooth surface, the Hilbert scheme of the surface is stratified according to the length of the intersection with the curve. The strata are highly singular. We show that this stratification admits a natural…
Let S be a smooth projective surface, and consider the following two subvarieties of the Hilbert scheme parameterizing closed subschemes of S of length n: A = {subschemes with support in a fixed point of S} B = {subschemes with support in…
In this paper, we give new explicit representations of the Hilbert scheme of $\mu$ points in $\PP^{r}$ as a projective subvariety of a Grassmanniann variety. This new explicit description of the Hilbert scheme is simpler than the previous…
We examine a sequence of examples of pairs of moduli spaces of sheaves on $\mathbb P^2$ where Le Potier's strange duality is expected to hold. One of the moduli spaces in these pairs is the Hilbert scheme of two points. We compute the…
We consider the following open questions. Fix a Hilbert function, $h$, that occurs for a reduced zero-dimensional subscheme of $\mathbb P^2$. Among all subschemes, $X$, with Hilbert function $h$, what are the possible Hilbert functions and…
Finite plane geometry is associated with finite dimensional Hilbert space. The association allows mapping of q-number Hilbert space observables to the c-number formalism of quantum mechanics in phase space. The mapped entities reflect…
Let $S$ be a finite set of points in the plane and let $\mathcal{T}(S)$ be the set of intersection points between pairs of lines passing through any two points in $S$. We characterize all configurations of points $S$ such that iteration of…
Let k be an algebraically closed field of characteristic p>2. By a result of Kumar and Thomsen, the standard Frobenius splitting of the affine plane induces a Frobenius splitting of the Hilbert scheme of n points in the plane. In this…
In this paper we investigate some algebraic and geometric consequences which arise from an extremal bound on the Hilbert function of the general hyperplane section of a variety (Green's Hyperplane Restriction Theorem). These geometric…
We study double line structures in projective spaces and quadric hypersurfaces, and investigate the geometry of irreducible components of Hilbert scheme of curves and moduli of stable sheaves of pure dimension 1 on a smooth quadric…
Over the last two decades, pseudospectral methods based on Lagrange interpolants have flourished in solving trajectory optimization problems and their flight implementations. In a seemingly unjustified departure from these highly successful…