Related papers: A discretized Severi-type theorem with application…

The Shafarevich conjecture/problem is about the finiteness of isomorphism classes of a family of varieties defined over a number field with good reduction outside a finite collection of places. For K3 surfaces, such a finiteness result was…

We prove that all minimal symplectic four-manifolds are essentially irreducible. We also clarify the relationship between holomorphic and symplectic minimality of K\"ahler surfaces. This leads to a new proof of the deformation-invariance of…

We develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in Lorentzian manifolds, where one has a lower bound on the Ricci tensor along timelike curves, and an upper bound on…

We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type we show that…

Tropical geometry yields good lower bounds, in terms of certain combinatorial-polyhedral optimisation problems, on the dimensions of secant varieties. In particular, it gives an attractive pictorial proof of the theorem of Hirschowitz that…

Let $K\subset\mathbb R^d$ be a compact subset equipped with a $\delta$-Ahlfors regular measure $\mu$. For any $\tau>1/d$ and any ``inhomogeneous'' vector $\boldsymbol{\theta}\in\mathbb R^d$, let $W_d(\psi_\tau,\boldsymbol{\theta})$ denote…

Let $ X = \Gamma\setminus \mathbb{H} $ be a non-elementary geometrically finite hyperbolic surface and let $ \delta $ denote the Hausdorff dimension of the limit set $ \Lambda(\Gamma) $. We prove that for every $ \varepsilon > 0 $ the…

For $0<k<1$, a finite-type $k$-surface in $3$-dimensional hyperbolic space is a complete, immersed surface of finite area and of constant extrinsic curvature equal to $k$. In [32], we showed that such surfaces have finite genus and finitely…

The Separating Hyperplane theorem is a fundamental result in Convex Geometry with myriad applications. Our first result, Random Separating Hyperplane Theorem (RSH), is a strengthening of this for polytopes. $\rsh$ asserts that if the…

Let $(\Sigma,p)$ be a pointed Riemann surface of genus $g\geq 1$. For any integer $k\geq 1$, we parametrize the space of meromorphic quadratic differentials on $\Sigma$ with a pole of order $(k+2)$ at $p$, having a connected critical graph…

The classical Besicovitch-Federer projection theorem implies that the d-dimensional Hausdorff measure of a set in Euclidean space with non-negligible d-unrectifiable part will strictly decrease under orthogonal projection onto almost every…

We consider, under suitable assumptions, the following situation: $\mathcal B$ is a component of the moduli space of polarized surfaces and $\mathcal V_{m,\delta}$ is the universal Severi variety over $\mathcal B$ parametrizing pairs…

The present work constitutes the third installment in a series of investigations devoted to discrete conformal structures on surfaces with boundary. In our preceding works \cite{X-Z DCS1, X-Z DCS2}, we established, respectively, a…

For $0<\alpha<1$ let $V(\alpha)$ denote the supremum of the numbers $v$ such that every $\alpha$-H\"older continuous function is of bounded variation on a set of Hausdorff dimension $v$. Kahane and Katznelson (2009) proved the estimate $1/2…

A projective hypersurface $X \subseteq \mathbb P^n$ has defect if $h^i(X) \neq h^i(\mathbb P^n)$ for some $i \in \{n, \dots, 2n-2\}$ in a suitable cohomology theory. This occurs for example when $X \subseteq \mathbb P^4$ is not $\mathbb…

We prove some sharp isoperimetric type inequalities for domains with smooth boundary on Riemannian manifolds. For example, using generalized convexity, we show that among all domains with a lower bound $l$ for the cut distance and Ricci…

We compute some numerical invariants of the lines on hyperplane sections of a smooth cubic threefold over complex numbers. We also prove that for any smooth hypersurface $X\subset \mathbb P^{n+1}$ of degree $d$ over an algebraically closed…

In 1985 Joe Harris proved the long standing claim of Severi that equisingular families of nodal plane curves are irreducible whenever they are non-empty. For families with more complicated singularities this is no longer true. Given a…

Let $X$ be a surface of general type with maximal Albanese dimension: if $K_X^2<\frac{9}{2}\chi(\mathcal{O}_X)$, one has $K_X^2\geq 4\chi(\mathcal{O}_X)+4(q-2)$. We give a complete classification of surfaces for which equality holds for…

We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4-manifold with a non-trivial Seiberg-Witten invariant. These allow one, for example, to exactly compute the infimum…