Related papers: Minimal log regular models of hyperbolic curves ov…

We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group.…

Let $C$ be a hyperelliptic curve of genus $g\ge 2$ over a discrete valuation field $K$ with perfect residue field. We study the minimal Weierstrass models of $C$. When there is more than one such model, we find interesting properties on the…

Inspired by the log Gromov-Witten (or GW) theory of Gross-Siebert/Abramovich-Chen, we introduce a geometric notion of log J-holomorphic curve relative to a simple normal crossings symplectic divisor defined in [FMZ1]. Every such moduli…

We study minimal Lagrangian surfaces in the complex hyperbolic quadric. We show that minimality of a Lagrangian surface is characterized by a loop of flat connections, which yields an associated $\mathbb S^1$-family of isometric…

Let $C/K$ be a smooth plane quartic over a discrete valuation field. We characterize the type of reduction (i.e. smooth plane quartic, hyperelliptic genus 3 curve or bad) over $K$ in terms of the existence of a special plane quartic model…

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…

Let $C:f=0$ be a reduced curve in the complex projective plane. The minimal degree $mdr(f)$ of a Jacobian syzygy for $f$, which is the same as the minimal degree of a derivation killing $f$, is an important invariant of the curve $C$, for…

We classify real families of minimal degree rational curves that cover an embedded rational surface. A corollary is that if the projective closure of a smooth surface is not biregular isomorphic to the projective closure of the unit-sphere,…

We extend the work of Cremona, Fisher and Stoll on minimising genus one curves of degrees 2,3,4,5, to some of the other representations associated to genus one curves, as studied by Bhargava and Ho. Specifically we describe algorithms for…

The Castelnuovo-Mumford regularity of the Jacobian algebra and of the graded module of derivations associated to a general curve arrangement in the complex projective plane are studied. The key result is an addition-deletion type result,…

This paper investigates the regularity of Lipschitz solutions $u$ to the general two-dimensional equation $\text{div}(G(Du))=0$ with highly degenerate ellipticity. Just assuming strict monotonicity of the field $G$ and heavily relying on…

We investigate the geometry of correspondences between curves, and prove that correspondences over a non-Archimedean valued field have potentially stable reduction, generalising and strengthening results of Coleman and Liu. This yields a…

F.-H. Lin studied minimal graphs of the Dirichlet problem in the hyperbolic space and proved that any such minimal graph has the same global regularity as the boundary if the dimension of the minimal graph is even and that there is an…

We prove an existence theorem for good moduli spaces, and use it to construct the second flip in the log minimal model program for the moduli space of stable curves. In fact, our methods give a uniform, self-contained construction of the…

We study the structure of Jacobians of geometrically reduced curves over arbitrary (i. e., not necessarily perfect) fields. We show that, while such a group scheme cannot in general be decomposed into an affine and an Abelian part as over…

We consider the germ of a reduced curve, possibly reducible. F.Delgado de la Mata proved that such a curve is Gorenstein if and only if its semigroup of values is symmetrical. We extend here this symmetry property to any fractional ideal of…

We study hidden boundary trace regularity for two-dimensional hyperbolic equations with boundary degeneracy governed by $\mcA\vp=-\Div(A\nabla \vp)$, where $A=\diag(1,r^\al)$ and $\al\in(0,1)$. We establish well-posedness in weighted…

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…

This paper is a continuation of our previous work [21], where we have established that, for the second-order degenerate hyperbolic equation (\p_t^2-t^m\Delta_x)u=f(t,x,u), locally bounded, piecewise smooth solutions u(t,x) exist when the…

We consider stable minimal surfaces of genus 1 in Euclidean space and in Riemannian manifolds. Under the condition of covering stability (all finite covers are stable) we show that a genus 1 finite total curvature minimal surface in…