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We study moduli stabilization for type IIB orientifolds compactified on Calabi-Yau threefolds in the region close to conifold singularities in the complex structure moduli space. The form of the periods implies new phenomena like…
We prove a formula for the ${\mathbb S}_n$-equivariant Euler characteristic of the moduli space of graphs $\mathcal{MG}_{g,n}$. Moreover, we prove that the rational ${\mathbb S}_n$-invariant cohomology of $\mathcal{MG}_{g,n}$ stabilizes for…
For a rational map $\phi: X \to G$ from a normal algebraic variety $X$ to a commutative algebraic group $G$, we define the modulus of $\phi$ as an effective divisor on $X$. We study the properties of the modulus. This work generalizes the…
We determine all of lines in the moduli space $M$ of stable bundles for arbitrary rank and degree. A further application of minimal rational curves is also given in last section.
We first describe the local and global moduli spaces of germs of foliations defined by analytic functions in two variables with p transverse smooth branches, and with integral multiplicities (in the univalued holomorphic case) or complex…
In this paper we propose two guiding principles that suggest a number of conjectures (some now proved) about various forms of rigidity for moduli spaces arising in algebraic geometry. Such conjectures have group-theoretic, topological and…
We investigate the moduli stabilization in string gas compactification. We first present a numerical evidence showing the stability of the radion and the dilaton. To understand this numerical result, we construct the 4-dimensional effective…
Fix a ruled surface S obtained as the projective completion of a line bundle L on a complex elliptic curve; we study the moduli problem of parametrizing certain pairs consisting of a sheaf E on S and a map of E to a fixed reference sheaf on…
The parameter space of $n$ ordered points in projective $d$-space that lie on a rational normal curve admits a natural compactification by taking the Zariski closure in $(\mathbb{P}^d)^n$. The resulting variety was used to study the…
We describe partial semi-simplicial resolutions of moduli spaces of surfaces with tangential structure. This allows us to prove a homological stability theorem for these moduli spaces, which often improves the known stability ranges and…
We study the Torelli morphism from the moduli space of stable curves to the moduli space of principally polarized stable semi-abelic pairs. We give two characterizations of its fibers, describe its injectivity locus, and give a sharp upper…
Two sets of conditions are presented for the compactness of a real plane algebraic curve, one sufficient and one necessary, in terms of the Newton polygon of the defining polynomial.
In this expository note, we offer an overview of the relationship between Hodge-theoretic and geometric compactifications of moduli spaces of algebraic varieties.
We use the theory of logarithmic line bundles to construct compactifications of spaces of roots of a line bundle on a family of curves, generalising work of a number of authors. This runs via a study of the torsion in the tropical and…
Let $(C,\underline{w})$ be a polarized nodal reducible curve. In this paper we consider coherent systems of type $(r,d,k)$ on $C$ with $k < r$. We prove that the moduli spaces of $(\underline{w},\alpha)$-stable coherent systems stabilize…
We present stability conditions for the category of coherent systems on an integral curve. We define a three-parameter family of pre-stability conditions in its derived category using tilting, and we then investigate when these conditions…
We show that if A is an abelian category satisfying certain mild conditions, then one can introduce the concept of a moduli space of (semi)stable objects which has the structure of a projective algebraic variety. This idea is applied to…
A sandpile is a cellular automaton on a graph that evolves by the following toppling rule: if the number of grains at a vertex is at least its valency, then this vertex sends one grain to each of its neighbors. In the study of pattern…
We develop the theory of adequate moduli spaces in characteristic $p$ (and mixed characteristic) characterizing quotients by geometrically reductive group schemes.
We introduce the notion of resultant of two planar curves in the tropical geometry framework. We prove that the tropicalization of the algebraic resultant can be used to compute the stable intersection of two tropical plane curves. It is…