Related papers: On smooth plane models for modular curves of Shimu…

Recently, the first author classified finite groups obtained as automorphism groups of smooth plane curves of degree $d \ge 4$ into five types. He gave an upper bound of the order of the automorphism group for each types. For one of them,…

We enumerate the singular algebraic curves in a complete linear system on a smooth projective surface. The system must be suitably ample in a rather precise sense. The curves may have up to eight nodes, or a triple point of a given type and…

We show that the moduli space of positive Ricci curvature metrics on all the total spaces of $S^7$-bundles over $S^8$ which are rational homology spheres has infinitely many path components. Furthermore, we carry out the diffeomorphism…

This paper is motivated by the real symplectic isotopy problem : does there exists a nonsingular real pseudoholomorphic curve not isotopic in the projective plane to any real algebraic curve of the same degree? Here, we focus our study on…

The global geometry of the moduli spaces of higher spin curves and their birational classification is largely unknown for g >= 2 and r > 2. Using quite related geometric constructions, we almost complete the picture of the known results in…

Let $E$ be a $\mathbb Q$-curve without complex multiplication. We address the problem of deciding whether $E$ is geometrically isomorphic to a strongly modular $\mathbb Q$-curve. We show that the question has a positive answer if and only…

We prove that integral points can be effectively determined on all but finitely many modular curves, and on all but one modular curve of prime power level.

In this paper we classify normal non--cyclic triple covers of $\bbP^2$ with branch curve of degree at most 10.

We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have…

Given two irreducible curves of the plane which have isomorphic complements, it is natural to ask whether there exists an automorphism of the plane that sends one curve on the other. This question has a positive answer for a large family of…

We show that for every positive integer n there is a simple closed curve in the plane (which can be taken infinitely differentiable and convex) which has exactly n inscribed squares.

When a non-singular complex projective surface $X$ satisfies that $K_X\sim 0$, we shall show that there are only finitely many isomorphic classes as abstract schemes in the set of moduli scheme of $H$-semistable sheaves with fixed Chern…

We prove an effective version of the Shafarevich conjecture (as proven by Faltings) for smooth quartic curves. To do so, we establish an effective version of Scholl's finiteness result for smooth del Pezzo surfaces of degree at most four.

We prove that the moduli space of curves with level structures has an enormous amount of rational cohomology in its cohomological dimension. As an application, we prove that the coherent cohomological dimension of the moduli space of curves…

We define the type of a plane curve as the initial degree of the corresponding Bourbaki ideal. Then we show that this invariant behaves well with respect to the union of curves. Curves of type $0$ are precisely the free curves, while curves…

The group PGL(3) of linear transformations of the projective plane acts naturally on the projective space parametrizing curves of a given degree. In this note we begin the study of the orbits of smooth curves under this action: we construct…

Some new results on plane F_{q^2}-maximal curves are stated and proved. It is known that the degree d of such curves is upper bounded by q+1 and that d=q+1 if and only if the curve is F_{q^2}-isomorphic to the Hermitian. We show that d\le…

In this paper we compute the degree of a curve which is the image of a mapping $z\longmapsto (f(z): g(z): h(z))$ constructed out of three linearly independent modular forms of the same even weight $\ge 4$ into $\mathbb P^2$. We prove that…

We prove the existence of integral canonical models of Shimura varieties of preabelian type with respect to primes of characteristic at least 5.

We recall two basic conjectures on the developables of convex projective curves, prove one of them and disprove the other in the firdt nontrivial case of curves in RP^3. Namely, we show that i) the tangent developable surface of any convex…