Related papers: Towards the weighted bounded negativity conjecture…
The smooth equimultiple locus of embedded algebroid surfaces appears naturally in many resolution process, both classical and modern. In this paper we explore how it changes by blowing--up.
The goal of this paper is to motivate a boundedness conjecture on nearby slopes of $\ell$-adic sheaves in positive characteristic, and to prove it for smooth curves. For a constructible $\ell$-adic sheaf, we prove the finiteness of the set…
Oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses. By self-intersection number one means the…
We provide a bound for $m$ such that the zero locus of a very general section of an $m$-multiple of some ample line bundle on a weighted projective space with isolated singularities is algebraically hyperbolic.
For an oriented surface $S$, the singular set of a fold map $f:S\rightarrow \mathbb{R}^2$ is a collection of smooth curves, also known as fold singularities. We construct a sharp lower bound on the number of self-intersections of such fold…
The minimal surfaces meeting in triples with equal angles along a common boundary naturally arise from soap films and other physical phenomenon. They are also the natural extension of the usual minimal surface. In this paper, we consider…
We provide a theoretical study of Algebraic Geometry codes constructed from abelian surfaces defined over finite fields. We give a general bound on their minimum distance and we investigate how this estimation can be sharpened under the…
The Klein and Wiman configurations are highly symmetric configurations of lines in the projective plane arising from complex reflection groups. One noteworthy property of these configurations is that all the singularities of the…
We prove a general fusion theorem for complete orientable minimal surfaces in $\mathbb{R}^3$ with finite total curvature. As a consequence, complete orientable minimal surfaces of weak finite total curvature with exotic geometry are…
Solutions to scalar curvature equations have the property that all possible blow-up points are isolated, at least in low dimensions. This property is commonly used as the first step in the proofs of compactness. We show that this result…
We construct an invariant of parametrized generic real algebraic surfaces in RP^3 which generalizes the Brown invariant of immersed surfaces from smooth topology. The invariant is constructed using the self intersection, which is a real…
We study the bounded negativity conjecture for non-quaternionic Hilbert modular surfaces and give an explicit bound for the special case of Hirzebruch-Zagier curves on Hilbert modular surfaces.
A viable and still unproved conjecture states that, if $X$ is a smooth algebraic surface and $C$ is a smooth algebraic curve in $X$, then $C$ realizes the smallest possible genus amongst all smoothly embedded $2$-manifolds in its homology…
In an earlier paper (D. S. Keeler, D. Rogalski, and J. T. Stafford, ``Naive noncommutative blowing up,'' Duke Math. J., 126 (2005), 491-546), we defined and investigated the properties of the naive blowup of an integral projective scheme X…
We investigate the crossover between weak and strong self-avoidance in a simulation of random surfaces with extrinsic curvature. We consider both dynamically triangulated and rigid surfaces with the two possible discretizations of the…
Recent literature on Weil-Petersson random hyperbolic surfaces has met a consistent obstacle: the necessity to condition the model, prohibiting certain rare geometric patterns (which we call tangles), such as short closed geodesics or…
In the long paper "Family Blowup formula, Admissible Graphs and the Enumeration of Singular Curves (I)" (appearing in JDG), the author solved the enumeration problem of nodal (or general singular) curve counting on algebraic surfaces by…
We investigate the limit behaviour of sequences of free boundary minimal hypersurfaces with bounded index and volume, by presenting a detailed blow-up analysis near the points where curvature concentration occurs. Thereby, we derive a…
We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let $k$ be a field and let $\mathcal{L}$ be a collection of $n$ space curves in $k^3$, with…
The doubling conjecture predicts that a manifold admits positive scalar curvature with mean convex boundary if and only if its double admits positive scalar curvature. We show that it holds true for manifolds where the inclusion of the…