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The geometry of the moduli space of stable spin curves is studied, with emphasis on its combinatorial properties. In this context, the standard graph theoretic framework is not just a book-keeping device: some purely combinatorial results…

The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to…

The aim of this paper is to study some modular contractions of the moduli space of stable pointed curves. These new moduli spaces, which are modular compactifications of the moduli space of smooth pointed curves, are related with the…

In 2010, Turaev introduced knotoids as a variation on knots that replaces the embedding of a circle with the embedding of a closed interval with two endpoints which here we call poles. We define generalized knotoids to allow arbitrarily…

In this paper, we survey some Galois-theoretic techniques for studying torsion points on curves. In particular, we give new proofs of some results of A. Tamagawa and the present authors for studying torsion points on curves with "ordinary…

We introduce a notion of limit linear series for nodal curves which are not of compact type. We give a construction of a moduli space of limit linear series, which works also in smoothing families, and we prove a corresponding…

We introduce circular evolutes and involutes of framed curves in the Euclidean space. Circular evolutes of framed curves stem from the curvature circles of Bishop directions and singular value sets of normal surfaces of Bishop directions.…

We introduce the notion of a generalized intersection pairing for an Artin stack with a proper good moduli space and nonempty stable part. For the moduli stack of semistable bundles over a smooth projective curve, there are four known…

In this paper we study the local geometry of the stack of pointed $A_r$-stable curves. In particular, we analyze the deformation theory of $A_r$-stable curves and their automorphism groups in order to study the combinatorics of families of…

Simple drawings are drawings of graphs in which the edges are Jordan arcs and each pair of edges share at most one point (a proper crossing or a common endpoint). We introduce a special kind of simple drawings that we call generalized…

We present a novel, log-radius profile representation for convex curves and define a new operation for combining the shape features of curves. Unlike the standard, angle profile-based methods, this operation accurately combines the shape…

The Torelli theorem establishes that the Jacobian of a smooth projective curve, together with the polarization provided by the theta divisor, fully characterizes the curve. In the case of nodal curves, there exists a concept known as fine…

The goal of this article is to motivate and describe how Gromov-Witten theory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from Gromov-Witten theory have led to both conjectures and…

This article treats various aspects of the geometry of the moduli of r-spin curves and its compactification. Generalized spin curves, or r-spin curves, are a natural generalization of 2-spin curves (algebraic curves with a…

Trigonal curves provide an example of Brill-Noether special curves. Theorem 1.3 of [9] characterizes the Brill-Noether theory of general trigonal curves and the refined stratification by Brill-Noether splitting loci, which parametrize line…

In the present paper, we study a new kind of anabelian phenomenon concerning the smooth pointed stable curves in positive characteristic. It shows that the topological structures of moduli spaces of curves can be understood from the…

The goal of this paper is to give a general theory of logarithmic Gromov-Witten invariants. This gives a vast generalization of the theory of relative Gromov-Witten invariants introduced by Li-Ruan, Ionel-Parker, and Jun Li, and completes a…

We construct new "virtually smooth" modular compactifications of spaces of maps from nonsingular curves to smooth projective toric varieties. They generalize Givental's compactifications, when the complex structure of the curve is allowed…

We calculate the cohomology spaces of the Hilbert schemes of points on surfaces with values in locally constant systems. For that purpose, we generalise I. Grojnoswki's and H. Nakajima's description of the ordinary cohomology in terms of a…

We consider the Alexander polynomial of a plane algebraic curve twisted by a linear representation. We show that it divides the product of the polynomials of the singularity links, for unitary representations. Moreover, their quotient is…