Related papers: Looijenga's Conjecture via Integral-affine Geometr…

Let X be the quotient of a smooth projective variety over a field by a finite group action (in which case we say X is pseudo-smooth), such that the singularities of X are isolated k-rational points. Let Y be obtained by blowing up these…

The purpose of this paper is to give an application of the gluing theorem for special Lagrangian submanifolds of a Calabi-Yau 3-fold. We proved a gluing theorem before to smooth a codimension-two singularity of a particular special…

Starting from an Enriques surface over $\mathbb{Q}(t)$ considered by Lafon, we give the first examples of smooth projective weakly special threefolds which fibre over the projective line in Enriques surfaces (resp. K3 surfaces) with nowhere…

We prove the Kawaguchi-Silverman conjecture (KSC), about the equality of arithmetic degree and dynamical degree, for every surjective endomorphism of any (possibly singular) projective surface. In high dimensions, we show that KSC holds for…

Along cuspidal edge singularities on a given surface in Euclidean 3-space, which can be parametrized by a regular space curve, a unit normal vector field $\nu$ is well-defined as a smooth vector field of the surface. A cuspidal edge…

We consider singularities of frontal surfaces of corank one and finite frontal codimension. We look at the classification under left-right-equivalence and introduce the notion of frontalisation for singularities of fold type. We define the…

The main result of the present paper is the proof of the Strange Duality for elliptic surfaces -- a duality between global sections of determinantal line bundles on moduli spaces of stable sheaves on a fixed elliptic surface. For this, we…

In real algebraic geometry, Lojasiewicz's theorem asserts that any integral curve of the gradient flow of an analytic function that has an accumulation point has a unique limit. Lojasiewicz proved this result in the early 1960s as a…

Let $X$ be a smooth threefold over an algebraically closed field of positive characteristic. We prove that an arbitrary flop of $X$ is smooth. To this end, we study Gorenstein curves of genus one and two-dimensional elliptic singularities…

A product-quotient surface is the minimal resolution of the singularities of the quotient of a product of two curves by the action of a finite group acting separately on the two factors. We classify all minimal product-quotient surfaces of…

We study deformations of rational curves and their singularities in positive characteristic. We use this to prove that if a smooth and proper surface in positive characteristic $p$ is dominated by a family of rational curves such that one…

An I-surface $S$ is an algebraic surface of general type with $K_S^2 = 1$ and $p_g(S) = 2$. Recent research has centered on trying to give an explicit description of the KSBA compactification of the moduli space of these surfaces. The…

We investigate the correspondence between three theories of deformations of rational surface singularities: de Jong and van Straten's picture deformations, Koll\'ar's P-resolutions, and Pinkham's smoothings of negative weights. We provide…

The sandwiched surface singularities are those rational surface singularities which dominate birationally smooth surface singularities. de Jong and van Straten showed that one can reduce the study of the deformations of a sandwiched surface…

We give more or less explicit equations for all two-dimensional cusp singularities of embedding dimension at least 4. They are closely related to Felix Klein's equations for universal curves with level n structure. The main technical result…

In this paper we propose and prove an anticyclotomic Iwasawa main conjecture for Heegner points on supersingular elliptic curves with $a_p=0$. The result has a "$\pm$" nature in the sense of Kobayashi. The proof uses a recent work of the…

We study equisingular deformation problems for curves and surfaces in algebraic families, with particular emphasis on situations where nodal behavior is no longer generic. Extending classical Severi theory, we develop deformation--theoretic…

The tractability conjecture for finite domain Constraint Satisfaction Problems (CSPs) stated that such CSPs are solvable in polynomial time whenever there is no natural reduction, in some precise technical sense, from the 3-SAT problem;…

In this paper we prove that if S is a smooth, irreducible, projective, rational, complex surface and D an effective, connected, reduced divisor on S, then the pair (S,D) is contractible if the log-Kodaira dimension of the pair is $-\infty$.…

We introduce a direct generalization of the Weinstein conjecture to closed, Lichnerowicz exact, locally conformally symplectic manifolds, (for short $\lcs$ manifolds). This conjectures existence of certain 2-curves in the manifold, which we…