Related papers: Smooth Herzog projective curves

We define a family of arithmetic zero cycles in the arithmetic Chow group of a modular curve X_0(N), for N>3 odd and squarefree, and identify the arithmetic degrees of these cycles as q-coefficients of the central derivative of a Siegel…

We classify minimal complex surfaces of general type with $p_g=q=3$. More precisely, we show that such a surface is either the symmetric product of a curve of genus 3 or a free $\Z_2-$quotient of the product of a curve of genus 2 and a…

Francesco Severi showed that equisingular families of plane nodal curves are T-smooth, i.e. smooth of the expected dimension, whenever they are non-empty. For families with more complicated singularities this is no longer true. Given a…

Let S be a Dedekind scheme with field of functions K. We show that if X_K is a smooth connected proper curve of positive genus over K, then it admits a N\'eron model over S, i.e., a smooth separated model of finite type satisfying the usual…

We study genus one curves of degree 5 defined by Pfaffians. We give new formulae for the invariants, and prove the equivalence of two different definitions of minimality. As an application we show that transformations between models with…

We prove that any non-commutative smooth projective variety with a Bridgeland stability condition of dimension less than $\frac{6}{5}$ must be a smooth projective curve. As a consequence, we deduce the non-existence of such categories with…

We provide new results and new proofs of results about the torsion of curves in $\mathbb{R}^3$. Let $\gamma$ be a smooth curve in $\mathbb{R}^3$ that is the graph over a simple closed curve in $\mathbb{R}^2$ with positive curvature. We give…

This research announcement discusses our results on Gromov-Witten theory of root gerbes. A complete calculation of genus 0 Gromov-Witten theory of $\mu_{r}$-root gerbes over a smooth base scheme is obtained by a direct analysis of virtual…

We construct a modular desingularisation of $\overline{\mathcal{M}}_{2,n}(\mathbb{P}^r,d)^{\text{main}}$. The geometry of Gorenstein singularities of genus two leads us to consider maps from prestable admissible covers: with this enhanced…

A conjecture by Corti, Filip and Petracci, inspired by mirror symmetry, states that smoothing types of affine Gorenstein toric 3-folds correspond to zero mutable Laurent polynomials. We propose a method to prove this conjecture via log…

This note describes the subring of the Grothendieck ring of k-varieties generated by smooth conics; finding many zero divisors. The proof uses only elementary projective geometry.

We define Gromov-Witten classes and invariants of smooth proper tame Deligne-Mumford stacks of finite presentation over a Dedekind domain. We prove that they are deformation invariants and verify the fundamental axioms. For a smooth proper…

We extend results of Chang and Ran regarding large dimensional families of immersed curves of positive genus in projective space in two directions. In one direction, we prove a sharp bound for the dimension of a complete family of smooth…

The paper concerns discrete versions of the three well-known results of projective differential geometry: the four vertex theorem, the six affine vertex theorem and the Ghys theorem on four zeroes of the Schwarzian derivative. We study…

Surfaces of general type with geometric genus $p_g=0$, which can be given as Galois covering of the projective plane branched over an arrangement of lines with Galois group $G=(\mathbb Z/q\mathbb Z)^k$, where $k\geq 2$ and $q$ is a prime…

We show the existence of toric resolution tower for an irreducible curve singularity which is explicitly described by Tschirnhausen polynomials. We deduce for a smooth affine plane curve from its topology restrictions for its singularity at…

Oort has conjectured that there do not exist Shimura curves contained generically in the Torelli locus of genus-$g$ curves when $g$ is large enough. In this paper we prove the Oort conjecture for Shimura curves of Mumford type and Shimura…

In this work, it is established that for a generic projective hypersurface $H\subset\mathbb{P}^n(\mathbb{C})$ of degree $d\geq(5n)^2\,n^{n}$, any holomorphic entire curve $f\colon\mathbb{C}\to\mathbb{P}^n(\mathbb{C})\setminus H$ has its…

We find some equivalences of the derived category of coherent sheaves on a Gorenstein genus one curve that preserve the (semi)-stability of pure dimensional sheaves. Using them we establish new identifications between certain Simpson moduli…

We define an integer-valued invariant of special cube complexes called the genus, and prove that having genus one characterizes special cube complexes with abelian fundamental group. Using the genus, we obtain a new proof that the…