Related papers: Ineffective descent of genus one curves

In this note we focus on the defect of singular plane curve that was recently introduced by Dimca. Roughly speaking, the defect of a reduced plane curve measures the discrepancy from the property of being a free curve. We find some…

In this paper, we study the deformation theory of degenerate algebraic curves on singular varieties which appear as the degenerate limit of families of varieties. For this purpose, we systematically develop a new method to calculate the…

The gonality conjecture predicts that the gonality of a curve can be read off Koszul cohomology of line bundles of sufficiently large degree. We verify this conjecture for generic curves of odd genus. The even-genus case was previously…

We give large families of Shimura curves defined by congruence conditions, all of whose twists lack $p$-adic points for some $p$. For each such curve we give analytically large families of counterexamples to the Hasse principle via the…

Strong consistency of the Bayes estimator of a regression curve for the $L^1$-squared loss function is proved. It is also shown the convergence to 0 of the Bayes risk of this estimator both for the $L^1$ and $L^1$-squared loss functions.…

We verify Green's conjecture for a generic $k$-gonal curve of genus $g$, for $g\geq k(k-1)/2$.

This work considers the effect of averaging, and more generally extrapolation, of the iterates of gradient descent in smooth convex optimization. After running the method, rather than reporting the final iterate, one can report either a…

We give a short and elementary proof of the $\ell^{2}$ decoupling inequality for the moment curve in $\mathbb{R}^k$, using a bilinear approach inspired by the nested efficient congruencing argument of Wooley (arXiv:1708.01220).

We develop a new method for establishing the extremality in the closed cone of effective curves on the moduli space of curves and determine the extremality of many boundary $1$-strata. As a consequence, by using a general criterion for…

Globally irreducible nodes (i.e. nodes whose branches belong to the same irreducible component) have mild effects on the most common topological invariants of an algebraic curve. In other words, adding a globally irreducible node (simple…

We revisit the nef curve cone theorem and use it to reprove the pseudo-effectivity of the second Chern classes for terminal weak Q-Fano varieties.

Base on a conjecture, we prove that for any smooth separated stack of finite type over a number field, its descent obstruction equals its iterated descent obstruction. As a consequence, we show that for any algebraic stack over a number…

We study the stable pairs theory of local curves in 3-folds with descendent insertions. The rationality of the partition function of descendent invariants is established for the full local curve geometry (equivariant with respect to the…

In this work, we mainly study the one-loop effective action for real scalar theories in non-homogeneous backgrounds in odd dimensions. It is shown that through the method studied in Ref. [1], it is possible to obtain a unified result for…

Let $C$ and $D$ be smooth, proper and geometrically integral curves over a finite field $F$. Any morphism from $D$ to $C$ induces a morphism of their \'etale fundamental groups. The anabelian philosophy proposed by Grothendieck suggests…

Let $F$ be a field complete with respect to a discrete valuation whose residue field is perfect of characteristic $p>0$. We prove that every smooth, projective, geometrically irreducible curve of genus one defined over $F$ with a non-zero…

The Castelnuovo bound conjecture, which is proposed by physicists, predicts an effective vanishing result for Gopakumar-Vafa invariants of Calabi-Yau 3-folds of Picard number one. Previously, it is only known for a few cases and all the…

In this paper we study the arithmetic and invariant theory of genus one normal curves embedded in $\mathbb{P}^{n-1}$. We generalize the notion of genus one model of degree $n$, introduced by Cremona, Fisher and Stoll for $n \leq 5$, to…

In this article we consider smooth projective curves $C$ of genus two described by integral equations of the form $y^2=xh(x)$, where $h(x)\in\mathbb{Z}[x]$ is monic of degree $4$. It turns out that if $h(x)$ is reducible, then the absolute…

Previous results on genera g of F_{q^2}-maximal curves are improved: (1) Either g\leq (q^2-q+4)/6, or g=\lfloor(q-1)^2/4\rfloor, or g=q(q-1)/2; (2) The hypothesis on the existence of a particular Weierstrass point in \cite{at} is proved;…