Related papers: Irreducibility and singularities of some nested Hi…

Given a finite collection $\mathbf{V}:=(V_1,\dots,V_N)$ of closed linear subspaces of a real Hilbert space $H$, let $P_i$ denote the orthogonal projection operator onto $V_i$ and $P_{i,\lambda}:= (1-\lambda)I + \lambda P_i$ denote its…

Starting with the irreducible triangulations of a fixed surface and splitting vertices, all the triangulations of the surface up to a given number of vertices can be generated. The irreducible triangulations have previously been determined…

We use generalised Bialynicki-Birula decomposition, apolarity and obstruction theories to prove non-reducedness of the Hilbert scheme of $13$ points on $\mathbb{A}^6$. Our argument doesn't involve computer calculations and gives an example…

In this paper, we give three bases for the cohomology groups of the Hilbert scheme of two points on projective space. Then, we use these bases to compute all effective and nef cones of higher codimensional cycles on the Hilbert scheme.…

We consider problems concerning the largest degrees of irreducible characters of symmetric groups, and the multiplicities of character degrees of symmetric groups. Using evidence from computer experiments, we posit several new conjectures…

Let $k$ be an algebraically closed field and let $\Hilb_{d}^{G}(\p{N})$ be the open locus of the Hilbert scheme $\Hilb_{d}(\p{N})$ corresponding to Gorenstein subschemes. We prove that $\Hilb_{d}^{G}(\p{N})$ is irreducible for $d\le9$, we…

We express nested Hilbert schemes of points and curves on a smooth projective surface as "virtual resolutions" of degeneracy loci of maps of vector bundles on smooth ambient spaces. We show how to modify the resulting obstruction theories…

We prove that a submaximal plane curve (i.e., an irreducible counterexample to Nagata's conjecture) with r singular points has sequence of multiplicities (m, n, ..., n) with m<sn for every integer with ((s-1)(s+2))^2 > 6.76(r-1).

In order to calculate the multiplicity of an isolated rational curve C in a local complete intersection variety X, i.e. the length of the Hilbert scheme of X at [C], it is important to study infinitesimal neighborhoods of the curve in X.…

We give an explicit stratification of the punctual Hilbert schemes of $n$ points of $\mathbb{A}^{m+1}$ with respect to $m$-dimensional partitions in the Grothendieck group of varieties. As an application, we calculate the classes of the…

When $s\ge k\ge 3$ and $n_1,\ldots ,n_k$ are large natural numbers, denote by $A_{s,k}(\mathbf n)$ the number of solutions in non-negative integers $\mathbf x$ to the system \[ x_1^j+\ldots +x_s^j=n_j\quad (1\le j\le k). \] Under…

Let $n$ be an integer$\geqq0$. Let $S^{n+2}_1$ (respectively, $S^{n+2}_2$) be the $(n+2)$-sphere embedded in the $(n+4)$-sphere $S^{n+4}$. Let $S^{n+2}_1$ and $S^{n+2}_2$ intersect transversely. Suppose that the smooth submanifold,…

We fix the lexicographic order $\prec$ on the polynomial ring $S=k[x_{1},...,x_{n}]$ over a ring $k$. We define $\Hi^{\prec\Delta}_{S/k}$, the moduli space of reduced Gr\"obner bases with a given finite standard set $\Delta$, and its open…

Over an infinite field $K$ with $\mathrm{char}(K)\neq 2,3$, we investigate smoothable Gorenstein $K$-points in a punctual Hilbert scheme from a new point of view, which is based on properties of double-generic initial ideals and of marked…

Let f(t,X) be an irreducible polynomial over the field of rational functions k(t), where k is a number field. Let O be the ring of integers of k. Hilbert's irreducibility theorem gives infinitely many integral specializations of t to values…

We work out details of the extrinsic geometry for two Hilbert schemes of some contemporary interest: the Hilbert scheme of two points on the projective plane and the dense open set parametrizing non-planar clusters in the punctual Hilbert…

Let $\mathcal{I}_{d,g,R}$ be the union of irreducible components of the Hilbert scheme whose general points parametrize smooth, irreducible, curves of degree $d$, genus $g$, which are non--degenerate in the projective space $\mathbb{P}^R$.…

To any unramified double cover $\pi:\tilde C \to C$ of projective irreducible and nonsingular curves one associates the Prym variety $P = P(\pi)$. For $C$ nonhyperelliptic of genus $g \geq 6$ we consider the natural embedding $\tilde C…

Given a map of vector bundles on a smooth variety, consider the deepest degeneracy locus where its rank is smallest. We show it carries a natural perfect obstruction theory whose virtual cycle can be calculated by the Thom-Porteous formula.…

It is well known that $n$-dimensional projective group gives rise to a non-homogenous representation of the Lie algebra $sl(n+1)$ on the polynomial functions of the projective space. Using Shen's mixed product for Witt algebras (also known…