Curves Of Genus Research Articles

Hodge conjecture for the moduli space of semi-stable sheaves over a nodal curve

In this article, we prove the Hodge conjecture for a desingularization of the moduli space of rank 2, semi-stable, torsion-free sheaves with fixed odd degree determinant over a very general irreducible nodal curve of genus at least 2. We also compute the algebraic Poincaré polynomial of the associated cohomology ring.

The local-global principle for integral points on stacky curves

We construct a stacky curve of genus 1 / 2 1/2 (i.e., Euler characteristic 1 1 ) over Z \mathbb {Z} that has an R \mathbb {R} -point and a Z p \mathbb {Z}_p -point for every prime p p but no Z \mathbb {Z} -point. This is best possible: we also prove that any stacky curve of genus less than 1 / 2 1/2 over a ring of S S -integers of a global field satisfies the local-global principle for integral points.

Globally generated vector bundles on and the unirationality of

Chang and Ran proved, in 1984, the unirationality of the moduli spaces of curves of genus 11, 12, and 13. They consider a family U of nonsingular space curves. If U is non-empty, it dominates the moduli space. Considering certain vector bundles related to the curves, one shows that U is unirational. Finally, Chang and Ran prove that U is non-empty by constructing smoothable reducible curves. We prove, instead, the existence of (almost) globally generated vector bundles of the above kind. The dependency locus of (rank − 1) general global sections of such a bundle is a nonsingular curve from U.

Singular intersections of subgroups and character varieties

We prove a global local rigidity result for character varieties of 3-manifolds into \(\mathrm {SL}_2({\mathbb {C}})\). Given a 3-manifold with toric boundary M satisfying some technical hypotheses, we prove that all but a finite number of its Dehn fillings \(M_{p/q}\) are globally locally rigid in the following sense: every irreducible representation \(\rho :\pi _1(M_{p/q})\rightarrow \mathrm {SL}_2({\mathbb {C}})\) is infinitesimally rigid, meaning that \(H^1(M_{p/q},\text {Ad}_\rho )=0\). This question arose from the study of asymptotics problems in topological quantum field theory developed in Charles and Marché (Knot state asymptotics II. Irreducible representations and the Witten conjecture, 2011). The proof relies heavily on recent progress in diophantine geometry and raises new questions of Zilber–Pink type. The main step is to show that a generic curve lying in a plane multiplicative torus intersects transversally almost all subtori of codimension 1. We prove an effective result of this form, based mainly on a height upper bound of Habegger.

Development of analytical seismic fragility functions for the common buildings in Iran

One of the main components for developing regional seismic risk models is the fragility functions of common building types. Due to the differences between the national design codes, construction practices, and construction materials, it is necessary to develop specific fragility functions for the common building types which are constructed in each region. For this reason, the present study is devoted to filling this essential gap for Iran. Therefore, in the first step, the existing building types in the country are classified into 31 categories regarding material, lateral-load-resisting system, height, and code level. Also, by conducting comprehensive studies on all previously performed research in the country, structural and dynamic parameters have been collected for buildings in each class. This information was used to compute a large set of backbone curves for Iranian buildings taxonomy. Then nearly three hundred appropriate fragility functions were generated using non-linear time-history analyses on the generic backbone curves and a large set of ground motion records. Based on the existing empirical data from past earthquakes in the country, the validation of the resulting fragility functions was carried out. The resulted fragility functions can be utilized in seismic risk assessment studies in the country.

Restricted Tangent Bundles for General Free Rational Curves

Abstract Suppose that $X$ is a smooth projective variety and that $C$ is a general member of a family of free rational curves on $X$. We prove several statements showing that the Harder–Narasimhan filtration of $T_{X}|_{C}$ is approximately the same as the restriction of the Harder–Narasimhan filtration of $T_{X}$ with respect to the class of $C$. When $X$ is a Fano variety, we prove that the set of all restricted tangent bundles for general free rational curves is controlled by a finite set of data. We then apply our work to analyze Peyre’s “freeness” formulation of Manin’s Conjecture in the setting of rational curves.

The Sato-Tate conjecture and Nagao's conjecture

Nagao's conjecture relates the rank of an elliptic surface to a limit formula arising from a weighted average of fibral Frobenius traces, and it is further generalized to smooth irreducible projective surfaces. We show that the Sato-Tate conjecture based on the random matrix model implies Nagao's conjecture for surfaces which form families of quadratic twists of fixed elliptic curves or fixed hyperelliptic curves of genus g≥1.

On algebraic curves with many automorphisms in characteristic p

Let \({\mathcal {X}}\) be an irreducible, non-singular, algebraic curve defined over a field of odd characteristic p. Let g and \(\gamma \) be the genus and p-rank of \({\mathcal {X}}\), respectively. The influence of g and \(\gamma \) on the automorphism group \(Aut({\mathcal {X}})\) of \({\mathcal {X}}\) is well-known in the literature. If \(g \ge 2\) then \(Aut({\mathcal {X}})\) is a finite group, and unless \({\mathcal {X}}\) is the so-called Hermitian curve, its order is upper bounded by a polynomial in g of degree four (Stichtenoth). In 1978 Henn proposed a refinement of Stichtenoth’s bound of degree 3 in g up to few exceptions, all having p-rank zero. In this paper a further refinement of Henn’s result is proposed. First, we prove that if an algebraic curve of genus \(g \ge 2\) has more than \(336g^2\) automorphisms then its automorphism group has exactly two short orbits, one tame and one non-tame, that is, the action of the group is completely known. Finally when \(|Aut({\mathcal {X}})| \ge 900g^2\) sufficient conditions for \({\mathcal {X}}\) to have p-rank zero are provided.

Rational points on algebraic curves in infinite towers of number fields

We study a natural question in the Iwasawa theory of algebraic curves of genus \(>1\). Fix a prime number p. Let X be a smooth, projective, geometrically irreducible curve defined over a number field K of genus \(g>1\), such that the Jacobian of X has good ordinary reduction at the primes above p. Fix an odd prime p and for any integer \(n>1\), let \(K_n^{(p)}\) denote the degree-\(p^n\) extension of K contained in \(K(\mu _{p^{\infty }})\). We prove explicit results for the growth of \(\#X(K_n^{(p)})\) as \(n\rightarrow \infty \). When the Jacobian of X has rank zero and the associated adelic Galois representation has big image, we prove an explicit condition under which \(X(K_{n}^{(p)})=X(K)\) for all n. This condition is illustrated through examples. We also prove a generalization of Imai’s theorem that applies to abelian varieties over arbitrary pro-p extensions.

Differential invariants of curves in G2 flag varieties

We compute the algebra of differential invariants of unparametrized curves in the homogeneous G2 flag varieties, namely in G2/P. This gives a solution to the equivalence problem for such curves. We consider the cases of integral and generic curves and relate the equivalence problems for all three choices of the parabolic subgroup P.

Perfect points on curves of genus 1 and consequences for supersingular K3 surfaces

We describe a method to show that certain elliptic surfaces do not admit purely inseparable multisections (equivalently, that genus 1 curves over function fields admit no points over the perfect closure of the base field) and use it to show that any non-Jacobian elliptic structure on a very general supersingular K3 surface has no purely inseparable multisections. We also describe specific examples of genus 1 fibrations on supersingular K3 surfaces without purely inseparable multisections.

Geometry of Genus One Fine Compactified Universal Jacobians

Abstract We introduce a general abstract notion of fine compactified Jacobian for nodal curves of arbitrary genus. We focus on genus $1$ and prove combinatorial classification results for fine compactified Jacobians in the case of a single nodal curve and in the case of the universal family $\overline {{\mathcal {C}}}_{1,n} / \overline {{\mathcal {M}}}_{1,n}$ over the moduli space of stable pointed curves. We show that if the fine compactified Jacobian of a nodal curve of genus $1$ can be extended to a smoothing of the curve, then it can be described as the moduli space of stable sheaves with respect to some polarisation. In the universal case we construct new examples of genus $1$ fine compactified universal Jacobians. Then we give a formula for the Hodge and Betti numbers of each genus $1$ fine compactified universal Jacobian $\overline {{\mathcal {J}}}^{d}_{1,n}$ and prove that their even cohomology is algebraic.

On the singular locus of a plane projection of a complete intersection

In this paper, we compute the number of self-intersections of a plane projection of a generic complete intersection curve defined by polynomials with given support. Moreover, we discuss the tropical counterpart of this problem.

On different expressions for invariants of hyperelliptic curves of genus 3

In this paper we give a passage formula between different invariants of genus 3 hyperelliptic curves: in particular between Tsuyumine and Shioda invariants. This is needed to get modular expressions for Shioda invariants, that is, for example, useful for proving the correctness of numerically computed equations of CM genus 3 hyperelliptic curves. On the other hand, we also get Shioda invariants described in terms of differences of roots of the equation defining the hyperelliptic curve, that has applications for studying the reduction type of the curve. Under certain conditions on its Jacobian, we give a criterion for determining the type of bad reduction of a genus 3 hyperelliptic curve.

Anabelian geometry of certain types of hyperbolic polycurves

In this paper, we study higher dimensional anabelian geometry. We prove the Grothendieck conjecture for certain types of hyperbolic polycurves (i.e., algebraic varieties in the form of successive fibrations by hyperbolic curves) over a sub-p-adic field (i.e., a field isomorphic to a subfield of a field finitely generated over the p-adic field Qp) for some prime number p. Among other things, we prove the Grothendieck conjecture for hyperbolic polycurves obtained as certain types of finite étale covers of the moduli space of curves of genus two with ordered r marked points over a sub-p-adic field.

Discriminative Shape Feature Pooling in Deep Neural Networks.

Although deep learning approaches are able to generate generic image features from massive labeled data, discriminative handcrafted features still have advantages in providing explicit domain knowledge and reflecting intuitive visual understanding. Much of the existing research focuses on integrating both handcrafted features and deep networks to leverage the benefits. However, the issues of parameter quality have not been effectively solved in existing applications of handcrafted features in deep networks. In this research, we propose a method that enriches deep network features by utilizing the injected discriminative shape features (generic edge tokens and curve partitioning points) to adjust the network’s internal parameter update process. Thus, the modified neural networks are trained under the guidance of specific domain knowledge, and they are able to generate image representations that incorporate the benefits from both handcrafted and deep learned features. The comparative experiments were performed on several benchmark datasets. The experimental results confirmed our method works well on both large and small training datasets. Additionally, compared with existing models using either handcrafted features or deep network representations, our method not only improves the corresponding performance, but also reduces the computational costs.

Hyperbolic secant varieties of M-curves

Abstract We relate the geometry of curves to the notion of hyperbolicity in real algebraic geometry. A hyperbolic variety is a real algebraic variety that (in particular) admits a real fibered morphism to a projective space whose dimension is equal to the dimension of the variety. We study hyperbolic varieties with a special interest in the case of hypersurfaces that admit a real algebraic ruling. The central part of the paper is concerned with secant varieties of real algebraic curves where the real locus has the maximal number of connected components, which is determined by the genus of the curve. For elliptic normal curves, we further obtain definite symmetric determinantal representations for the hyperbolic secant hypersurfaces, which implies the existence of symmetric Ulrich sheaves of rank one on these hypersurfaces. We also use this to derive better bounds on the size of semidefinite representations for convex hulls of real algebraic curves of genus 1.

On birational Torelli theorems

Let G be a simple simply-connected connected linear algebraic group over \({\mathbb {C}}\). We prove a 2-birational Torelli theorem for the moduli space of semistable principal G-bundles over a smooth curve of genus \(\ge 3\), which says that if two such moduli spaces are 2-birational, then the curves are isomorphic. We also prove a 3-birational Torelli theorem for the moduli space of stable symplectic parabolic bundles over a smooth curve of genus \(\ge 4\).

Counting and Cutting Rich Lenses in Arrangements of Circles

We show that the maximum number of pairwise nonoverlapping $k$-rich lenses (lenses formed by at least $k$ circles) in an arrangement of $n$ circles in the plane is $O(n^{3/2}\log(n / k^3)/k^{5/2} + n/k)$, and the sum of the degrees of the lenses of such a family (where the degree of a lens is the number of circles that form it) is $O(n^{3/2}\log(n/k^3)/k^{3/2} + n)$. Two independent proofs of these bounds are given, each interesting in its own right (so we believe). The second proof gives a bound that is weaker by a polylogarithmic factor. We then show that these bounds lead to the known bound of Agarwal et al. [J. ACM, 51 (2004), pp. 139--186] and Marcus and Tardos [J. Combin. Theory Ser. A, 113 (2006), pp. 675--691] on the number of point-circle incidences in the plane. Extensions to families of more general algebraic curves and some other related problems are also considered.

𝔸1–connected components of ruled surfaces

A conjecture of Morel asserts that the sheaf of $\mathbb A^1$-connected components of a space is $\mathbb A^1$-invariant. Using purely algebro-geometric methods, we determine the sheaf of $\mathbb A^1$-connected components of a smooth projective surface, which is birationally ruled over a curve of genus $>0$. As a consequence, we show that Morel's conjecture holds for all smooth projective surfaces over an algebraically closed field of characteristic $0$.