Related papers: A higher genus circle method and an application to…

A degree $d$ genus $g$ cover of the complex projective line by a smooth curve $C$ yields a vector bundle on the projective line by pushforward of the structure sheaf. Which bundles are possible? Equivalently, which…

We study the density of rational points on a higher-dimensional orbifold $(\mathbb{P}^{n-1},D)$ when $D$ is a $\mathbb{Q}$-divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational…

For smoothly bounded, strongly $\mathbb{C}$-convex domains, one can use the Fefferman form or its variants to define projectively invariant norms on sections of holomorphic line bundles, producing a Hardy space. In two variables, we…

We develop new techniques to study regularity questions for moduli spaces of pseudoholomorphic curves that are multiply covered. Among the main results, we show that unbranched multiple covers of closed holomorphic curves are generically…

In this paper we connect degenerations of Fano threefolds by projections. Using Mirror Symmetry we transfer these connections to the side of Landau--Ginzburg models. Based on that we suggest a generalization of Kawamata's categorical…

We describe a practical algorithm for computing Brauer-Manin obstructions to the existence of rational points on hyperelliptic curves defined over number fields. This offers advantages over descent based methods in that its correctness does…

Let S be a smooth projective surface equipped with a line bundle H. Lehn's conjecture is a formula for the top Segre class of the tautological bundle associated to H on the Hilbert scheme of points of S. Voisin has recently reduced Lehn's…

Let $f\colon C \rightarrow \mathbb{P}^1$ be a degree $k$ genus $g$ cover. The stratification of line bundles $L \in \mathrm{Pic}^d(C)$ by the splitting type of $f_*L$ is a refinement of the stratification by Brill-Noether loci $W^r_d(C)$.…

Let n > 2 and let d < (n+1)/2. We prove that for a general hypersurface X of degree d in P^n, all the genus 0 Kontsevich moduli spaces M_{0,n}(X,e) are irreducible, reduced, local complete intersection stacks of the expected dimension.

We study the summands of the decomposition theorem for the Hitchin system for $\mathrm{GL}_n$, in arbitrary degree, over the locus of reduced spectral curves. A key ingredient is a new correspondence between these summands and the topology…

We describe a method to prove new integral inequalities for stable minimal hypersurfaces in Euclidean space. As an application, we give a simple proof that complete, two sided, stable minimal hypersurfaces in $\mathbb{R}^4$ are hyperplanes.…

We sharpen and generalize the dimension growth bounds for the number of points of bounded height lying on an irreducible algebraic variety of degree $d$, over any global field. In particular, we focus on the affine hypersurface situation by…

We give two proofs that appropriately defined congruence subgroups of the mapping class group of a surface with punctures/boundary have enormous amounts of rational cohomology in their virtual cohomological dimension. In particular we give…

We introduce a new method to study mixed characteristic deformation of line bundles. In particular, for sufficiently large smooth projective families $f : \mathscr{X} \to \mathscr{S}$ defined over the ring of $N$-integers…

Many classical results in algebraic geometry arise from investigating some extremal behaviors that appear among projective varieties not lying on any hypersurface of fixed degree. We study two numerical invariants attached to such…

We study genus $g$ hyperelliptic curves with reduced automorphism group $A_5$ and give equations $y^2=f(x)$ for such curves in both cases where $f(x)$ is a decomposable polynomial in $x^2$ or $x^5$. For any fixed genus the locus of such…

We study parameter spaces of linear series on projective curves in the presence of unibranch singularities, i.e. {\it cusps}; and to do so, we stratify cusps according to value semigroup. We show that {\it generalized Severi varieties} of…

We study the stable hyperelliptic locus, i.e. the closure, in the Deligne- Mumford moduli space of stable curves, of the locus of smooth hyperelliptic curves. Working on a suitable blowup of the relative Hilbert scheme (of degree 2)…

We study the existence of a decomposition of the diagonal for bidegree hypersurfaces in a product of projective spaces. Using a cycle theoretic degeneration technique due to Lange, Pavic and Schreieder, we develop an inductive procedure…

We prove that a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group admits a self-map of absolute degree greater than one if and only if it is virtually trivial. This generalizes in every dimension the…