Related papers: Towards a Combinatorial Description of the Interse…

We define higher categorical invariants (gerbes) of codimension two algebraic cycles and provide a categorical interpretation of the intersection of divisors on a smooth proper algebraic variety. This generalization of the classical…

Based on the combinatorial description of the moduli spaces of curves provided by Strebel differentials, Witten and Kontsevich have introduced combinatorial cohomology classes $W_{(m_0,m_1,m_2,\dots),n}$, and conjectured that these can be…

The Hilbert scheme of n points in the projective plane has a natural stratification obtained from the associated Hilbert series. In general, the precise inclusion relation between the closures of the strata is still unknown. Guerimand…

For a relative effective divisor $\mathcal{C}$ on a smooth projective family of surfaces $q:\mathcal{S}\rightarrow B$, we consider the locus in $B$ over which the fibres of $\mathcal{C}$ are $\delta$-nodal curves. We prove a conjecture by…

We study the existence of components with the expected number of moduli of the Hilbert scheme of integral nodal curves $C \subset \mathbb {P}^r$ with prescribed degree, arithmetic genus and number of singular points.

Let $X$ be a smooth polarized algebraic surface over the compex number field. We discuss the invariants obtained from the moduli stacks of semistable sheaves of arbitrary ranks on $X$. For that purpose, we construct the virtual fundamental…

We compute the Chow ring of an arbitrary heavy/light Hassett space $\bar{M}_{0, w}$. These spaces are moduli spaces of weighted pointed stable rational curves, where the associated weight vector $w$ consists of only heavy and light weights.…

We calculate the classes of the universal hyperelliptic Weierstrass divisor $\bar{{\mathcal H}}_{g,w}$ and the universal $g^1_2$ divisor $\bar{{\mathcal H}}_{g,g^1_2}$. Our results are expressed in terms of a basis for $Cl(\bar{{\mathcal…

Given two elliptic curves, each of which is associated with a projection map that identifies opposite elements with respect to the natural group structure, we investigate how their corresponding projective images of torsion points…

We categorify the localized intersection product on arithmetic schemes defined by Kato--Saito in \cite{katosaito04}. As an application, we prove a generalization of Bloch conductor conjecture.

On a smooth variety, Serre's intersection formula computes intersection multiplicities via an alternating sum of the lengths of Tor groups. When the variety is singular, the corresponding sum can be a divergent series. But there are…

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 present a series of new results we obtained recently about the intersection numbers of tautological classes on moduli spaces of curves, including a simple formula of the n-point functions for Witten's $\tau$ classes, an effective…

We give a new definition of higher arithmetic Chow groups for smooth projective varieties defined over a number field, which is similar to Gillet and Soul\'e's definition of arithmetic Chow groups. We also give a compact description of the…

We prove that any element in the Chow ring of the Hilbert scheme $\text{Hilb}_n$ of $n$ points on a smooth surface $S$ is a universal class, i.e. the pushforward of a polynomial in the Chern classes of the universal subscheme on…

Let $A$ be a closed affine subspace and let $B$ be a hyperplane in a Hilbert space. Suppose we are given their associated nearest point mappings $P_A$ and $P_B$, respectively. We present a formula for the projection onto their intersection…

We derive some explicit expressions for correlators on Grassmannian G_r(C^n) as well as on the moduli space of holomorphic maps, of a fixed degree d, from sphere into the Grassmannian. Correlators obtained on the Grassmannain are a first…

The equivariant and ordinary cohomology rings of Hilbert schemes of points on the minimal resolution C^2//G for cyclic G are studied using vertex operator technique, and connections between these rings and the class algebras of wreath…

We calculate the genus zero cobordism-valued Gromov-Witten invariants of a point by refining the string equation on $\overline{\mathcal{M}}_{0,n}$ from the Chow ring to algebraic cobordism. This gives inductive formulas for cobordism-valued…

We compute the Chow motive and the Chow groups with rational coefficients of the Hilbert scheme of points on a smooth algebraic surface