Related papers: Arithmetic genus of integral space curves

We establish an area-type formula for the intrinsic spherical Hausdorff measure of every regular curve embedded in an arbitrary graded group.

We compute the genus one family Gromov-Witten invariants of K3 surfaces for non-primitive classes. These calculations verify Gottsche-Yau-Zaslow formula for non-primitive classes with index two. Our approach is to use the genus two…

We provide an explicit and algorithmic version of a theorem of Momose classifying isogenies of prime degree of elliptic curves over number fields, which we implement in Sage and PARI/GP. Combining this algorithm with recent work of…

In this paper we study the geometry of the Severi varieties parametrizing curves on the rational ruled surface $\fn$. We compute the number of such curves through the appropriate number of fixed general points on $\fn$, and the number of…

We show variants of the genus inequality for the irreducible components of the special fiber of an arithmetic curve over a henselian discrete valuation ring of residue characteristic zero that take into account the non-existence of…

Associated with a prime homology class $\beta \in P_2(X,\Z)$ (i.e. $\beta=p\alpha$ and $\alpha \in H_2(X,\Z)$ imply $p=1$ or $p$ is an odd prime) on a symplectic three-manifold with vanishing first Chern class, we count the embedded…

As another application of the degeneration methods of [V3], we count the number of irreducible degree $d$ geometric genus $g$ plane curves, with fixed multiple points on a conic $E$, not containing $E$, through an appropriate number of…

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…

We consider the locus of irreducible nonsingular rational curves of degree d Pn, n>2, meeting a generic collection of linear subspaces. When this locus is 0 (resp 1)- dimensional, we compute (recursively) its degree (resp. geometric genus).…

In this work we compute the first integral cohomology of the pure mapping class group of a non-orientable surface of infinite topological type and genus at least 3. To this purpose, we also prove several other results already known for…

We introduce orbifold Euler numbers for normal surfaces with Q-divisors. These numbers behave multiplicatively under finite maps and in the log canonical case we prove that they satisfy the Bogomolov-Miyaoka-Yau type inequality. As a…

The main goal of this article is to expand the theory of invariants of Artin-Schreier curves by giving a complete classification in genus 3 and 4. To achieve this goal, we first establish standard forms of Artin-Schreier curves and…

We prove a conjecture of Maulik, Pandharipande, and Thomas expressing the Gromov--Witten invariants of K3 surfaces for divisibility two curve classes in all genus in terms of weakly holomorphic quasimodular forms of level two. Then, we…

We give upper bounds on the genus of a curve with general moduli assuming that it can be embedded in a projective nonsingular surface $Y$ so that $\dim(|C|) > 0$. We find such bounds for all types of surfaces of intermediate Kodaira…

We establish upper and lower bounds for the number of integral points which lie within a neighbourhood of a smooth nondegenerate curve in $\mathbb{R}^n$ for $n\geq 3$. These estimates are new for $n\geq 4$, and we recover an earlier result…

Fix integers $r\geq 4$ and $i\geq 2$ (for $r=4$ assume $i\geq 3$). Assuming that the rational number $s$ defined by the equation $\binom{i+1}{2}s+(i+1)=\binom{r+i}{i}$ is an integer, we prove an upper bound for the genus of a reduced and…

I give a new proof, in scheme-theoretic language, of Tate's old result on genus-change over nonperfect fields in characteristic p>0. Namely, for normal geometrically integral curves, the difference between arithmetic and geometric genus…

Let X be a K3 surface with a primitive ample divisor H, and let $\beta=2[H]\in H_2(X, \mathbf Z)$. We calculate the Gromov-Witten type invariants $n_{\beta}$ by virtue of Euler numbers of some moduli spaces of stable sheaves. Eventually, it…

Let $K$ be a quadratic field which is not an imaginary quadratic field of class number one. We describe an algorithm to compute the primes $p$ for which there exists an elliptic curve over $K$ admitting a $K$-rational $p$-isogeny. This…

Hirzebruch surfaces provide an excellent example to underline the fact that in general symplectic manifolds, Gromov-Witten invariants might well count curves in the boundary components of the moduli space. We use this example to explain in…