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.

Nodal curves and polarizations with good properties

In this paper we deal with polarizations on a nodal curve C with smooth components. Our aim is to study and characterize a class of polarizations, which we call “good”, for which depth one sheaves on C reflect some properties that hold for vector bundles on smooth curves. We will concentrate, in particular, on the relation between the $${{\underline{w}}}$$ w ̲ -stability of $${\mathcal {O}}_C$$ O C and the goodness of $${{\underline{w}}}$$ w ̲ . We prove that these two concepts agree when C is of compact type and we conjecture that the same should hold for all nodal curves.

A Torelli type theorem for nodal curves

The moduli space of Gieseker vector bundles is a compactification of moduli of vector bundles on a nodal curve. This moduli space has only normal-crossing singularities and it provides flat degeneration of the moduli of vector bundles over a smooth projective curve. We prove a Torelli type theorem for a nodal curve using the moduli space of stable Gieseker vector bundles of fixed rank (strictly greater than [Formula: see text]) and fixed degree such that rank and degree are co-prime.

Two Moduli Spaces of Calabi–Yau type

We show $\overline{\mathcal{M}}_{10, 10}$ and $\overline{\mathcal{F}}_{11,9}$ have Kodaira dimension zero. Our method relies on the construction of a number of curves via nodal Lefschetz pencils on blown-up $K3$ surfaces. The construction further yields that any effective divisor in $\overline{\mathcal{M}}_{g}$ with slope $<6+(12-\delta )/(g+1)$ must contain the locus of curves that are the normalization of a $\delta $-nodal curve lying on a $K3$ surface of genus $g+\delta $.

Fourier–Mukai Transform on a Compactified Jacobian

We use Fourier–Mukai transform to compute the cohomology of the Picard bundles on the compactified Jacobian of an integral nodal curve $Y$. We prove that the transform gives an injective morphism from the moduli space of vector bundles of rank $r \ge 2$ and degree $d$ ($d$ sufficiently large) on $Y$ to the moduli space of vector bundles of a fixed rank and fixed Chern classes on the compactified Jacobian of $Y$. We show that this morphism induces a morphism from the moduli space of vector bundles of rank $r \ge 2$ and a fixed determinant of degree $d$ on $Y$ to the moduli space of vector bundles of a fixed rank with a fixed determinant and fixed Chern classes on the compactified Jacobian of $Y$.

Hilbert–Mumford criterion for nodal curves

We prove by the Hilbert–Mumford criterion that a slope stable polarized weighted pointed nodal curve is Chow asymptotic stable. This generalizes the result of Caporaso on stability of polarized nodal curves and of Hassett on weighted pointed stable curves polarized by the weighted dualizing sheaves. It also solves a question raised by Mumford and Gieseker, to prove the Chow asymptotic stability of stable nodal curves by the Hilbert–Mumford criterion.