Related papers: Brolin's Theorem for curves in two complex dimensi…

We settle the automorphism groups of curves appearing in a classification list of smooth plane curves with at least two Galois points. One of them is an ordinary curve whose automorphism group exceeds the Hurwitz bound.

In this paper we begin to study curves on a weighted projective plane with one trivial weight, ${\mathbb P}(1,m,n)$, by determining the genus of curves of Fermat type. These are curves defined by a ``homogeneous'' polynomial analagous to…

Using the theory of minimal models of quasi-projective surfaces we give a new proof of the theorem of Lin-Zaidenberg which says that every topologically contractible algebraic curve in the complex affine plane has equation $X^n=Y^m$ in some…

For a given complex projective variety, the existence of entire curves is strongly constrained by the positivity properties of the cotangent bundle. The Green-Griffiths-Lang conjecture stipulates that entire curves drawn on a variety of…

We provide sharp lower bounds for the multiplicity of a local holomorphic foliation defined in a complex surface in terms of data associated to a germ of invariant curve. Then we apply our methods to invariant curves whose branches are…

Via taking connected components of preimages, a Thurston map $f: (S^2, P_f) \to (S^2, P_f)$ induces a pullback relation on the set of isotopy classes of curves in the complement of its postcritical set $P_f$. We survey known results about…

A closed subscheme of codimension two $T \subset P^2$ is a quasi complete intersection (q.c.i.) of type $(a,b,c)$ if there exists a surjective morphism $\mathcal{O} (-a) \oplus \mathcal{O} (-b) \oplus \mathcal{O} (-c) \to \mathcal{I} _T$.…

Stack-theoretic blow-ups have proven to be efficient in resolving singularities over fields of characteristic zero. In this article, we move forward towards positive characteristic where new challenges arise. In particular, the dimension of…

Let X be a smooth projective hyperelliptic curve over an algeraically closed field k of prime characteristic p. The aim of this note is to find necessary and sufficient conditions on the automorphism group of the curve X to be lifted to…

We study complex projective plane curves with a given group of automorphisms. Let $G$ be a simple primitive subgroup of $PGL(3, \mathbb{C})$, which is isomorphic to $A_{6}$, $A_{5}$ or $PSL(2, \mathbb{F}_{7})$. We obtain a necessary and…

We prove an effective version of a theorem of Dufresnoy: For any set of 2n+1 hyperplanes in general position in n-dimensional complex projective space, we find an explicit constant K such that for every holomorphic map f from the unit disc…

Let X be a smooth projective curve of genus $g\geq 2$ defined over an algebraically closed field k of characteristic $p>0$ and let $F:X\rightarrow X_{1}$ be the relative k-linear Frobenius map. We prove (Theorem 1.1) E is a stable bundle on…

Let $\mathcal{C}$ be an irreducible plane curve of $\text{PG}(2,\mathbb{K})$ where $\mathbb{K}$ is an algebraically closed field of characteristic $p\geq 0$. A point $Q\in \mathcal{C}$ is an inner Galois point for $\mathcal{C}$ if the…

We establish a type of the Picard's theorem for entire curves in $P^n(\mathbb C)$ whose spherical derivative vanishes on the inverse images of hypersurface targets. Then, as a corollary, we prove that there is an union $D$ of finite number…

The space $\mathbf{H}^{4,2}$ of vectors of norm -1 in $\mathbb{R}^{4,3}$ has a natural pseudo-Riemannian metric and a compatible almost complex structure. The group of automorphisms of both of these structures is the split real form $G_2'$.…

A family of proper smooth curves of genus $\geq 2$, parametrised by an open dense subset $U$ of a normal variety $S$, extends to $S$ if the natural map $\pi_1(U) \to \pi_1(S)$ on fundamental groups is an isomorphism. The criterion of this…

We study the holonomy cocycle H of a holomorphic foliation \Fc by Riemann surfaces defined on a compact complex projective surface X satisfying the following two conditions: 1) its singularities E are all hyperbolic; 2) there is no…

Let E be a transitive Courant algebroid with scalar product of neutral signature. A generalized almost complex structure \mathcal J on E is a skew-symmetric smooth field of endomorphisms of E which squares to minus the identity. We say that…

In this note, we prove a criteria for supersingularity when the variety has a large automorphism group and a perfect bilinear pairing. This criteria unifies and extends many known results on the supersingularity of curves and varieties and…

In projective space over fields of characteristic different from 2, the normal bundle of a general nondegenerate rational curve is balanced. The corresponding statement for rational curves in other Grassmannians can fail. Nevertheless, we…