Geometrization of Trigonometric Solutions of the Associative and Classical Yang–Baxter Equations

We describe a geometric construction of all nondegenerate trigonometric solutions of the associative and classical Yang–Baxter equations. In the associative case, the solutions come from symmetric spherical orders over the irreducible nodal curve of arithmetic genus $1$, while in the Lie case they come from spherical sheaves of Lie algebras over the same curve.

Algebro-Geometric Solutions of the Coupled Chaffee-Infante Reaction Diffusion Hierarchy

The coupled Chaffee-Infante reaction diffusion (CCIRD) hierarchy associated with a 3 × 3 matrix spectral problem is derived by using two sets of the Lenard recursion gradients. Based on the characteristic polynomial of the Lax matrix for the CCIRD hierarchy, we introduce a trigonal curve K m − 2 of arithmetic genus m − 2 , from which the corresponding Baker-Akhiezer function and meromorphic functions on K m − 2 are constructed. Then, the CCIRD equations are decomposed into Dubrovin-type ordinary differential equations. Furthermore, the theory of the trigonal curve and the properties of the three kinds of Abel differentials are applied to obtain the explicit theta function representations of the Baker-Akhiezer function and the meromorphic functions. In particular, algebro-geometric solutions for the entire CCIRD hierarchy are obtained.

Moduli of stable sheaves supported on curves of genus three contained in a quadric surface

We study the moduli space of stable sheaves of Euler characteristic 1 supported on curves of arithmetic genus 3 contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers by studying the variation of the moduli spaces of α-semi-stable pairs. We classify the stable sheaves using locally free resolutions or extensions. We give a global description: the moduli space is obtained from a certain flag Hilbert scheme by performing two flips followed by a blow-down.

Algebro-geometric constructions of the Heisenberg hierarchy

The Heisenberg hierarchy and its Hamiltonian structure are derived respectively by virtue of the zero-curvature equation and the trace identity. With the help of the Lax matrix, we introduce an algebraic curve ${\mathcal{K}}_{n}$ of arithmetic genus n, from which we define meromorphic function ϕ and straighten out all of the flows associated with the Heisenberg hierarchy under the Abel–Jacobi coordinates. Finally, we achieve the explicit theta function representations of solutions for the whole Heisenberg hierarchy as a result of the asymptotic properties of ϕ.

Singular Curves of Low Degree and Multifiltrations from Osculating Spaces

In order to study projections of smooth curves, we introduce multifiltrations obtained by combining flags of osculating spaces. We classify all configurations of singularities occurring for a projection of a smooth curve embedded by a complete linear system away from a projective linear space of dimension at most two. In particular, we determine all configurations of singularities of non-degenerate degree $d$ rational curves in $\mathbb{P}^n$ when $d-n\leq 3$ and $d<2n$. Along the way, we describe the Schubert cycles giving rise to these projections. We also reprove a special case of the Castelnuovo bound using these multifiltrations: under the assumption $d<2n$, the arithmetic genus of any non-degenerate degree $d$ curve in $\mathbb{P}^n$ is at most $d-n$.

Severi varieties and Brill–Noether theory of curves on abelian surfaces

Severi varieties and Brill–Noether theory of curves on K3 surfaces are well understood. Yet, quite little is known for curves on abelian surfaces. Given a general abelian surface S with polarization L of type {(1,n)} , we prove nonemptiness and regularity of the Severi variety parametrizing δ-nodal curves in the linear system {|L|} for {0\leq\delta\leq n-1=p-2} (here p is the arithmetic genus of any curve in {|L|} ). We also show that a general genus g curve having as nodal model a hyperplane section of some {(1,n)} -polarized abelian surface admits only finitely many such models up to translation; moreover, any such model lies on finitely many {(1,n)} -polarized abelian surfaces. Under certain assumptions, a conjecture of Dedieu and Sernesi is proved concerning the possibility of deforming a genus g curve in S equigenerically to a nodal curve. The rest of the paper deals with the Brill–Noether theory of curves in {|L|} . It turns out that a general curve in {|L|} is Brill–Noether general. However, as soon as the Brill–Noether number is negative and some other inequalities are satisfied, the locus {|L|^{r}_{d}} of smooth curves in {|L|} possessing a {g^{r}_{d}} is nonempty and has a component of the expected dimension. As an application, we obtain the existence of a component of the Brill–Noether locus {{\mathcal{M}}^{r}_{p,d}} having the expected codimension in the moduli space of curves {{\mathcal{M}}_{p}} . For {r=1} , the results are generalized to nodal curves.

Local BPS Invariants: Enumerative Aspects and Wall-Crossing

We study the BPS invariants for local del Pezzo surfaces, which can be obtained as the signed Euler characteristic of the moduli spaces of stable one-dimensional sheaves on the surface $S$. We calculate the Poincaré polynomials of the moduli spaces for the curve classes $\beta $ having arithmetic genus at most 2. We formulate a conjecture that these Poincaré polynomials are divisible by the Poincaré polynomials of $((-K_S).\beta -1)$-dimensional projective space. This conjecture motivates the upcoming work on log BPS numbers [8].

Classification of degree two curves in the symmetric square with positive self-intersection

We give a precise classification of the pairs (C, B͠) with C a smooth curve of genus g and B͠ ⊂ C(2) a curve of degree two and positive self-intersection. We prove that there are no such pairs if g < pa(B͠) < 2g−1. We study the singularities and self-intersection of any degree two curve in C(2). Moreover, we give examples of curves with arithmetic genus in the Brill–Noether range and positive self-intersection on C × C.

MODULI OF CURVES WITH NONSPECIAL DIVISORS AND RELATIVE MODULI OF -STRUCTURES

In this paper, for each $n\geqslant g\geqslant 0$ we consider the moduli stack $\widetilde{{\mathcal{U}}}_{g,n}^{ns}$ of curves $(C,p_{1},\ldots ,p_{n},v_{1},\ldots ,v_{n})$ of arithmetic genus $g$ with $n$ smooth marked points $p_{i}$ and nonzero tangent vectors $v_{i}$ at them, such that the divisor $p_{1}+\cdots +p_{n}$ is nonspecial (has $h^{1}=0$ ) and ample. With some mild restrictions on the characteristic we show that it is a scheme, affine over the Grassmannian $G(n-g,n)$ . We also construct an isomorphism of $\widetilde{{\mathcal{U}}}_{g,n}^{ns}$ with a certain relative moduli of $A_{\infty }$ -structures (up to an equivalence) over a family of graded associative algebras parametrized by $G(n-g,n)$ .