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We show that the singularities of spacelike maximal surfaces in Lorentz-Minkowski 3-space generically consist of cuspidal edges, swallowtails and cuspidal cross caps. The same result holds for spacelike mean curvature one surfaces in de…
We solve the problem on flat extensions of a generic surface with boundary in Euclidean 3-space, relating it to the singularity theory of the envelope generated by the boundary. We give related results on Legendre surfaces with boundaries…
We consider the natural problem of counting isotopy classes of essential surfaces in 3-manifolds, focusing on closed essential surfaces in a broad class of hyperbolic 3-manifolds. Our main result is that the count of (possibly disconnected)…
Let $S$ be a nonsingular minimal complex projective surface of general type and the canonical map of $S$ is generically finite. Beauville showed that the geometric genus of the image of the canonical map is vanishing or equals the geometric…
The authors consider a scattering problem for electric potentials that have a component which is critically singular in the sense of Lebesgue spaces, and a component given by a measure supported on a compact Lipschitz hypersurface. They…
We construct a klt del Pezzo surface in characteristic three violating the Kawamata-Viehweg vanishing theorem. As a consequence we show that there exists a Kawamata log terminal threefold singularity which is not Cohen-Macaulay in…
On $(M,g)$ a compact riemannian $4-$manifold we consider the prescribed $Q-$curvature equation defined on $M$ with finite singular sources. We first prove a classification theorem for singular Liouville equations defined on $\mathbb R^4$…
We give a bound on the number of isolated, essential singularities of determinantal quartic surfaces in 3-space. We also provide examples of different configurations of real singularities on quartic surfaces with a definite Hermitian…
We establish estimates for the number of solutions of certain affine congruences. These estimates are then used to prove Manin's conjecture for a cubic surface split over Q and whose singularity type is D_4. This improves on a result of…
This paper contains examples of closed aspherical manifolds obtained as a by-product of recent work by the author [arXiv:math.GR/0509490] on the relative strict hyperbolization of polyhedra. The following is proved. (I) Any closed…
In this paper, we study the theory of complements, introduced by Shokurov, for Calabi-Yau type varieties with the coefficient set $[0,1]$. We show that there exists a finite set of positive integers $\mathcal{N}$, such that if a threefold…
This paper is motivated by two problems in the theory of Diophantine approximation, namely, Davenport's problem regarding badly approximable points on submanifolds of a Euclidean space and Schmidt's problem regarding the intersections of…
We investigate 3-dimensional almost Kenmotsu manifolds satisfying special types of nullity conditions depending on two smooth functions $\kappa,\mu$. When either $\kappa<-1$ and $\mu=0$ or $h=0$, such conditions coincide with the…
We show that if $x$ is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approximation at $x$, then $x$ is already a denting point. It turns…
The existence is proved of two new families of locally Cohen-Macaulay sextic threefolds in $\mathbb{P}^5$, which are not quadratically normal. These threefolds arise naturally in the realm of first order congruences of lines as focal loci…
We develop the deformation theory of Calabi-Yau threefolds, by which we mean 3-dimensional complex manifolds with a nowhere-vanishing holomorphic 3-form, on manifolds with boundary. The boundary data is a closed, real 3-form on the…
Triangulations and higher triangulations axiomatize the calculus of derived cokernels when applied to strings of composable morphisms. While there are no cubical versions of (higher) triangulations, in this paper we use coherent diagrams to…
In this paper, we construct counterexamples to the boundedness of generalised log canonical models of surfaces with fixed appropriate invariants, where the underlying varieties can have arbitrary Kodaira dimension. This answers a question…
This work presents a range of triangulated characterizations for important classes of singularities such as derived splinters, rational singularities, and Du Bois singularities. An invariant called 'level' in a triangulated category can be…
We investigate the existence of complete intersection threefolds $X \subset \mathbb{P}^n$ with only isolated, ordinary multiple points and we provide some sufficient conditions for their factoriality.