Related papers: Hirota Varieties and Rational Nodal Curves
We introduce a collection of nonlinear integrable partial differential-difference equations that are satisfied by the one-point distribution functions of some classical integrable KPZ models. Moreover, these equations can be regarded as…
Let V^{r}_{d,g, \delta} be the Hilbert scheme of nodal curves in P^r of degree d and arithmetic genus g with \delta nodes. Under suitable numerical assumptions on d and g, for every 0 \le \delta \le g we construct an irreducible component…
We study the family of irreducible curves with $\delta$ nodes belonging to a free linear system $|C|$ with smooth general member on a surface $S$ such that $|K_S|$ is ample. Under the assumption that $C$ is numerically equivalent to $pK_S$,…
We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\mathbb{P}^r$. In this…
For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the…
The motivic Hilbert zeta function of a variety is the generating function for classes in the Grothendieck ring of varieties of Hilbert schemes of points of the variety. In this paper, the motivic Hilbert zeta function of a reduced curve is…
In this work, we consider an integrable three-component coupled Hirota (tcCH) equations in detail via the Riemann-Hilbert (RH) approach. We present some properties of the spectral problems of the tcCH equations with $4\times4$ the Lax pair.…
The BKP hierarchy has a two-component analogue (the 2-BKP hierarchy). Dispersionless limit of this multi-component hierarchy is considered on the level of the $\tau$-function. The so called dispersionless Hirota equations are obtained from…
In this paper we determine the irreducible components of the Hilbert schemes H(4,g) of locally Cohen-Macaulay space curves of degree four and arbitrary arithmetic genus g. We show that these Hilbert schemes are connected, in spite of having…
We discuss the origin of the associativity (WDVV) equations in the context of quasiclassical or Whitham hierarchies. The associativity equations are shown to be encoded in the dispersionless limit of the Hirota equations for KP and Toda…
A conjecture by Yves Andre and Frans Oort says that closed subvarieties of Shimura varieties that contain a Zariski dense subset of special points are subvarieties of Hodge type. We prove this in the case where the subvariety is a curve…
We investigate space curves with large cohomology. To this end we introduce curves of subextremal type. This class includes all subextremal curves. Based on geometric and numerical characterizations of curves of subextremal type, we show…
We construct an integrable hierarchy in the form of Hirota quadratic equations (HQE) that governs the Gromov--Witten (GW) invariants of the Fano orbifold projective curve $\mathbb{P}^1_{a_1,a_2,a_3}$. The vertex operators in our…
We study the Hilbert scheme $\mathcal{H}^\mathcal{L}_{d,g,r}$ parametrizing smooth, irreducible, non-degenerate and linearly normal curves of degree $d$ and genus $g$ in $\mathbb{P}^r$ whose complete and very ample hyperplane linear series…
This is a short review of the construction of quasi-periodic (algebraic-geometrical) solutions to hierarchies of nonlinear integrable equations. As is well known, the solutions are expressed through Riemann's theta-functions associated with…
We consider a generalized Riemann-Hurwitz formula as it may be applied to rational maps between projective varieties having an indeterminacy set and fold-like singularities. The case of a holomorphic branched covering map is recalled. Then…
A parametrization of irreducible unitary representations associated with the regular adjoint orbits of a hyperspecial compact subgroup of a reductive group over a non-dyadic non-archimedean local filed is presented. The parametrization is…
We consider modular properties of nodal curves on general $K3$ surfaces. Let $\mathcal{K}_p$ be the moduli space of primitively polarized $K3$ surfaces $(S,L)$ of genus $p\geqslant 3$ and $\mathcal{V}_{p,m,\delta}\to \mathcal{K}_p$ be the…
We investigate limit linear series on chains of elliptic curves, giving a simple proof of a conjecture of Farkas stating the existence of curves with a theta-characteristic with a given number of sections for the expected range of genera.…
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…