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This paper addresses some conjectures and questions regarding the absolute and relative compactifications of the $\SL(2,\C)$-character variety of an $n$-punctured Riemann surface without boundary. We study a class of projective…
The paper contains a description of the links of complex surface germ.
In algebraic geometry, trigonal curves can always be embedded into Hirzebruch surfaces. In tropical geometry, the notion of trigonality does not have a unique translation. We focus on the characterization in terms of the existence of a…
In this paper we characterize the compact orbifolds, quotients $ X = \mathcal{D}/ \Gamma$ of a bounded symmetric domain $\mathcal{D}$ with no higher dimensional ball factor by the action of a discontinuous group $\Gamma$, as those…
A key property of an algebraic variety is whether it is absolutely irreducible, meaning that it remains irreducible over the algebraic closure of its defining field, and determining absolute irreducibility is important in algebraic geometry…
We show that a large square of sidelength $x$ can be packed by unit squares in a manner so that the wasted space $W(x) = O(x^{3/5})$.
We construct locally recoverable codes with hierarchy from surfaces in $\mathbb{A}^3$ admitting a fibration by curves of Artin-Schreier or Kummer type. We derive the parameters of our codes by leveraging the geometry and arithmetic of the…
The relationship between nonnegative polynomials and sums of squares is a classical topic in real algebraic geometry. We study \emph{stubborn polynomials} $f$ on a real variety $X$, which are polynomials nonnegative on $X$, such that no odd…
We develop a valuation-theoretic framework for studying tangent cones of torsion-free sheaves on algebraic varieties. To analyze these objects, we introduce a slope stability theory, including the Harder-Narasimhan filtrations, for finitely…
Nekrasov's gauge origami theory provides a (complex) 4-dimensional generalization of the ADHM quiver and its moduli spaces of representations. We describe the origami moduli space as the zero locus of an isotropic section of a quadratic…
We study the relative homology group of an affine hyperplane arrangement and its Poincar\'e dual, the cohomology at finite distance of the complement. We give an Orlik--Solomon-type description of the latter, and identify it with the vector…
We construct a normal projective $\mathbb{Q}$-Gorenstein surface over an algebraically closed field whose canonical ring is not finitely generated. Moreover, we provide a counterexample to the minimal model program for…
We generalize the formula for the log canonical threshold(LCT) of plane curves over the complex numbers to arbitrary characteristics. Our proof relies purely on valuation theory, instead of on the theory of $D$-modules.
We introduce an integral version of the Hodge polynomial, which encodes the integral cohomology of smooth projective varieties. We prove it extends to a function which is well-defined on the Grothendieck ring of varieties and we obtain as a…
Above a Laurent polynomial f one makes grow a vector space of vanishing cycles (after the work of Sabbah, singularity setting), a graded Milnor ring (after the work of Kouchnirenko) and an orbifold cohomology ring (after the work of…
Let $k$ be an algebraically closed field of characteristic $p >0$. We consider the variety of nilpotent pairs $(A,B)$ with $[A,B]=\lambda I$, namely the set of pairs $ X = \{ (A,B) \in M_n(k) \times M_n(k) \mid A,B \text{ nilpotent},…
We study compactifications of the moduli space of a plane cubic curve marked by \(n\) labeled points up to projective equivalence via Geometric Invariant Theory (GIT). Specifically, we provide a complete description of the GIT walls and…
A result of Teissier says that the cone over one of classical polygon examples in the real projective space gives, by complexification, a surface singularity which is not Whitney equisingular to a singularity defined over the field of…
We prove a Baum--Bott type residual formula for one-dimensional holomorphic foliations, and logarithmic along free divisors. More precisely, this provides a Baum--Bott theorem for a foliated triple $(X, \mathcal{F}, D)$, where $\mathcal{F}$…
We define categories of Stokes filtered and Stokes graded $G$-local systems for reductive groups $G$ and use the formalism of Tannakian categories to show that they are equivalent to the category of $G$-connections. We then use the…