Related papers: The Chow Ring of the Hilbert Cube

In previous papers the authors gave formulae for generating series of classes (in the Grothendieck ring of complex quasi-projective varieties) of Hilbert schemes of zero-dimensional subschemes on smooth varieties and on orbifolds in terms…

Let $K$ be an algebraically closed field of characteristic zero. Algebraic structures of a specific type (e.g. algebras or coalgebras) on a given vector space $W$ over $K$ can be encoded as points in an affine space $U(W)$. This space is…

We give a complete classification of quadratic algebras A, with Hilbert series $H_A=(1-t)^{-3}$, which is the Hilbert series of commutative polynomials on 3 variables. Koszul algebras as well as algebras with quadratic Gr\"obner basis among…

Let X be a smooth, connected, dimension n, quasi-projective variety imbedded in \PP_N. Consider integers {k_1,...,k_r}, with k_i>0, and the Hilbert Scheme H_{k_1,...,k_r}(X) of aligned, finite, degree \sum k_i, subschemes of X, with…

Let X be a normal projective variety admitting an action of a semisimple group with a unique closed orbit. We construct finitely many rational curves in X, all having a common point, such that every effective one-cycle on X is rationally…

The toric Hilbert scheme parametrizes all algebras isomorphic to a given semigroup algebra as a multigraded vectorspace. All components of the scheme are toric varieties, and among them, there is a fairly well understood coherent component.…

We prove that the Hilbert scheme of 11 points on a smooth threefold is irreducible. In the course of the proof, we present several known and new techniques for producing curves on the Hilbert scheme.

This paper proposes a conjectural picture for the structure of the Chow ring of a (projective) hyper-K\"ahler variety, and the construction of a Beauville decomposition, with emphasis on the Chow group of $0$-cycles, which is endowed with a…

We compute an explicit closed formula for the Hilbert polynomial of the Jacobian algebra $M(f)$ of a reduced surface $X:f=0$ in $\mathbb P^3$ in terms of the graded Betti numbers of the algebra $M(f)$. When $X$ has only isolated…

For an arbitrary field K, let I be an ideal in the ring K[[x,y]] expressible as a polynomial in either the pair of ideals (x, y^4) and (x,y) or the pair (x,y^3) and (x^2, y). Let G be the group of automorphisms of K[[x,y]] sending the ideal…

Motivated by two norm equations used to characterize the Friedrichs angle, this paper studies $C^*$-isomorphisms associated with two projections by introducing the matched triple and the semi-harmonious pair of projections. A triple…

A quantum theory in a finite-dimensional Hilbert space can be geometrically formulated as a proper Hamiltonian theory as explained in [2, 3, 7, 8]. From this point of view a quantum system can be described in a classical-like framework…

A generic K3 surface of degree 2t is a general complex projective K3 surface whose Picard group is generated by the class of an ample divisor whose with respect to the intersection form is 2t. We show that if X is the Hilbert square of a…

In this paper, we study the Gromov-Witten theory of the Hilbert schemes X^{[n]} of points on smooth projective surfaces X with positive geometric genus p_g. Using cosection localization technique due to Y. Kiem and J. Li [KL1, KL2], we…

Let $G_k$ be a split algebraic group over $k$. Given a non-trivial $G_k$-torsor $G$, we consider the (twisted) flag variety $F=G/B_k$ for the Borel subgroup $B_k$ containing in $G_k$. The purpose of this paper is the study how to compute…

Let $\mathcal{I}_{d,g,r}$ be the union of irreducible components of the Hilbert scheme whose general points correspond to smooth irreducible non-degenerate curves of degree $d$ and genus $g$ in $\mathbb{P}^r$. We use families of curves on…

We show that the orbifold Chow ring of a root stack over a well-formed weighted projective space can be naturally seen as the Jacobian algebra of a function on a singular variety given by a partial compactification of its Ginzburg-Landau…

For a very general polarized $K3$ surface $S\subset \mathbb{P}^g$ of genus $g\ge 5$, we study the linear system on the Hilbert square $S^{[2]}$ parametrizing quadrics in $\mathbb{P}^g$ that contain $S$. We prove its very ampleness for…

We give an explicit presentation of the integral Chow ring of a stack of smooth plane cubics. We also determine some relations in the general case of hypersurfaces of any dimension and degree.

We give a `geometrical' construction of an action of a Heisenberg algebra on the homology of the moduli spaces of torsion free sheaves on a complex smooth connected projective surface, framed along a smooth connected genus zero curve. This…