Smooth Projective Curve Of Genus Research Articles

A functorial approach to the stability of vector bundles

On a normal projective variety the locus of μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-stable bundles that remain μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-stable on all Galois covers prime to the characteristic is open in the moduli space of Gieseker semistable sheaves. On a smooth projective curve of genus at least 2 this locus is big in the moduli space of stable bundles. As an application we obtain a very different behaviour of the étale fundamental group in positive vs. characteristic 0.

Essentially finite G-torsors

Let X be a smooth projective curve of genus g, defined over an algebraically closed field k, and let G be a connected reductive group over k. We say that a G-torsor is essentially finite if it admits a reduction to a finite group, generalising the notion of essentially finite vector bundles to arbitrary groups G. We give a Tannakian interpretation of such torsors, and we prove that all essentially finite G-torsors have torsion degree, and that the degree is 0 if X is an elliptic curve. We then study the density of the set of k-points of essentially finite G-torsors of degree 0, denoted MGef,0, inside MGss,0, the k-points of all semistable degree 0 G-torsors. We show that when g=1, MGef⊂MGss,0 is dense. When g>1 and when char(k)=0, we show that for any reductive group of semisimple rank 1, MGef,0⊂MGss,0 is not dense.

Diagonal Dimension of Curves

Abstract We prove Conjecture 4.16 of the paper [ 2] of Elagin and Lunts; namely, that a smooth projective curve of genus at least $1$ over a field has diagonal dimension $2$.

Derived equivalence and fibrations over curves and surfaces

We prove that the bounded derived category of coherent sheaves on a smooth projective complex variety reconstructs the isomorphism classes of fibrations onto smooth projective curves of genus g≥2. Moreover, in dimension at most four, we prove that the same category reconstructs the isomorphism classes of fibrations onto normal projective surfaces with positive holomorphic Euler characteristic and admitting a finite morphism to an abelian variety. Finally, we study the derived invariance of a class of fibrations with minimal base dimension under the condition that all the Hodge numbers of type h0,p(X) are derived invariant.

Equivariant approach to weighted projective curves

We investigate group actions on the category of coherent sheaves over weighted projective lines. We show that the equivariant category with respect to certain finite group action is equivalent to the category of coherent sheaves over a weighted projective curve, where the genus of the underlying smooth projective curve and the weight data are given by explicit formulas. Moreover, we obtain a trichotomy result for all such equivariant relations.This equivariant approach provides an effective way to investigate smooth projective curves via weighted projective lines. As an application, we give a description for the category of coherent sheaves over a hyperelliptic curve of arbitrary genus. Links to smooth projective curves of genus 2 and also to Arnold's exceptional unimodal singularities are also established.

Fundamental group schemes of n-fold symmetric product of a smooth projective curve

Let k be an algebraically closed field of characteristic \(p > 0\). Let X be an irreducible smooth projective curve of genus g over k. Fix an integer \(n \ge 2\), and let \(S^n(X)\) be the n-fold symmetric product of X. In this article, we find the S-fundamental group scheme and Nori’s fundamental group scheme of \(S^n(X)\).

Surface braid groups, Finite Heisenberg covers and double Kodaira fibrations

We exhibit new examples of double Kodaira fibrations by using finite Galois covers of a product $\Sigma_b \times \Sigma_b$, where $\Sigma_b$ is a smooth projective curve of genus $b \geq 2$. Each cover is obtained by providing an explicit group epimorphism from the pure braid group $\mathsf{P}_2(\Sigma_b)$ to some finite Heisenberg group. In this way, we are able to show that every curve of genus $b$ is the base of a double Kodaira fibration; moreover, the number of pairwise non-isomorphic Kodaira fibred surfaces fibering over a fixed curve $\Sigma_b$ is at least $\boldsymbol{\omega}(b+1)$, where $\boldsymbol{\omega} \colon \mathbb{N} \to \mathbb{N}$ stands for the arithmetic function counting the number of distinct prime factors of a positive integer. As a particular case of our general construction, we obtain a real $4$-manifold of signature $144$ that can be realized as a real surface bundle over a surface of genus $2$, with fibre genus $325$, in two different ways. This provides (to our knowledge) the first "double" solution to a problem from Kirby's list in low-dimensional topology.

INTERSECTION COHOMOLOGY OF THE MODULI SPACE OF HIGGS BUNDLES ON A GENUS 2 CURVE

AbstractLetCbe a smooth projective curve of genus$2$. Following a method by O’Grady, we construct a semismall desingularisation$\tilde {\mathcal {M}}_{Dol}^G$of the moduli space$\mathcal {M}_{Dol}^G$of semistableG-Higgs bundles of degree 0 for$G=\mathrm {GL}(2,\mathbb {C}), \mathrm {SL}(2,\mathbb {C})$. By the decomposition theorem of Beilinson, Bernstein and Deligne, one can write the cohomology of$\tilde {\mathcal {M}}_{Dol}^G$as a direct sum of the intersection cohomology of$\mathcal {M}_{Dol}^G$plus other summands supported on the singular locus. We use this splitting to compute the intersection cohomology of$\mathcal {M}_{Dol}^G$and prove that the mixed Hodge structure on it is pure, in analogy with what happens to ordinary cohomology in the smooth case of coprime rank and degree.

Rational Points on Solvable Curves over ℚ via Non-Abelian Chabauty

Abstract We study the Selmer varieties of smooth projective curves of genus at least two defined over $\mathbb{Q}$ which geometrically dominate a curve with CM Jacobian. We extend a result of Coates and Kim to show that Kim’s non-abelian Chabauty method applies to such a curve. By combining this with results of Bogomolov–Tschinkel and Poonen on unramified correspondences, we deduce that any cover of P$^1$ with solvable Galois group, and in particular any superelliptic curve over $\mathbb{Q}$, has only finitely many rational points over $\mathbb{Q}$.

Genus 3 hyperelliptic curves with CM via Shimura reciprocity

Up to isomorphism over C, every simple principally polarized abelian variety of dimension 3 is the Jacobian of a smooth projective curve of genus 3. Furthermore, this curve is either a hyperelliptic curve or a plane quartic. Given a sextic CM field K, we show that if there exists a hyperelliptic Jacobian with CM by K, then all principally polarized abelian varieties that are Galois conjugated to it are hyperelliptic. Using Shimura's reciprocity law, we give an algorithm for computing approximations of the invariants of the initial curve, as well as their Galois conjugates. This allows us ton define and compute class polynomials for genus 3 hyperelliptic curves with CM.

Counting Higgs bundles and type quiver bundles

We prove a closed formula counting semistable twisted (or meromorphic) Higgs bundles of fixed rank and degree over a smooth projective curve of genus $g$ defined over a finite field, when the twisting line bundle degree is at least $2g-2$ (this includes the case of usual Higgs bundles). This yields a closed expression for the Donaldson–Thomas invariants of the moduli spaces of twisted Higgs bundles. We similarly deal with twisted quiver sheaves of type $A$ (finite or affine), obtaining in particular a Harder–Narasimhan-type formula counting semistable $U(p,q)$-Higgs bundles over a smooth projective curve defined over a finite field.

Automorphisms of relative Quot schemes

Let \(X\rightarrow S\) be a smooth family of projective curves over an algebraically closed field k of characteristic zero. Assume that both X and S are smooth projective varieties and let E be a vector bundle of rank r over X and \(\mathbb {P}(E)\) be its projectivization. Fix \(d\ge 1\). Let \(\mathcal {Q}(E,d)\) be the relative quot scheme of torsion quotients of E of degree d. Then we show that if \(r\ge 3\), then the identity component of the group of automorphisms of \(\mathcal {Q}(E,d)\) over S is isomorphic to the identity component of the group of automorphisms of \(\mathbb {P}(E)\) over S. We also show that under additional hypotheses, the same statement is true in the case when \(r=2\). As a corollary, the identity component of the automorphism group of flag scheme of filtrations of torsion quotients of \(\mathcal {O}^{r}_{C}\), where \(r\ge 3\) and C a smooth projective curve of genus \(\ge 2\) is computed.

ON MAXIMALLY FROBENIUS DESTABILISED VECTOR BUNDLES

Let $X$ be a smooth projective curve of genus $g\geq 2$ over an algebraically closed field $k$ of characteristic $p>0$. We show that for any integers $r$ and $d$ with $0<r<p$, there exists a maximally Frobenius destabilised stable vector bundle of rank $r$ and degree $d$ on $X$ if and only if $r\mid d$.

Statistics of moduli space of vector bundles

Let X be a smooth projective curve of genus g≥1 over a finite field Fq with q elements, such that the function field Fq(X) is a geometric Galois extension of the rational function field Fq(x) with N=#Gal(Fq(X)/Fq(x)). Let ML(2,1) be the moduli space of rank 2 stable vector bundles over X‾=X×FqFq‾ with fixed determinant L, where L is a line bundle on X‾ of degree 1. Let Nq(ML(2,1)) be the cardinality of the set of Fq-rational points of ML(2,1). We give an estimate of Nq(ML(2,1)) in terms of N,q and g. Further we study the fluctuations of the quantity logNq(ML(2,1))−3(g−1)logq as the curve X as well as L varies over a large family of hyperelliptic curves (N=2) of genus g≥2. We find the limiting distribution of Nq(ML(2,1))−3(g−1)log⁡q as g grows and q is fixed, in terms of its characteristic function. When both g and q grow we see that q3/2(log⁡Nq(ML(2,1))−3(g−1)log⁡q) has a standard Gaussian distribution.

On large theta-characteristics with prescribed vanishing

Let C be a smooth projective curve of genus $$g\ge 2$$ . Fix an integer $$r\ge 0$$ , and let $$\underline{k}=(k_1,\ldots ,k_n)$$ be a sequence of positive integers with $$\sum _{i=1}^n k_i =g-1$$ . In this paper, we study n-pointed curves $$(C,p_1,\ldots ,p_n)$$ such that the line bundle $$L:=O_C\left( \sum _{i=1}^n k_i p_i\right) $$ is a theta-characteristic with $$h^0\left( C,L\right) \ge r+1$$ and $$h^0\left( C,L\right) \equiv r+1 \,\mathrm {(mod\,2)}$$ . We prove that they describe a sublocus $${\mathcal {G}}^r_g(\underline{k})$$ of $${\mathcal {M}}_{g,n}$$ having codimension at most $$g-1+\frac{r(r-1)}{2}$$ . Moreover, for any $$r\ge 0$$ , $$\underline{k}$$ as above, and g greater than an explicit integer g(r) depending on r, we present irreducible components of $${\mathcal {G}}^r_g(\underline{k})$$ attaining the maximal codimension in $${\mathcal {M}}_{g,n}$$ , so that the bound turns out to be sharp.

Algebraic cycles and the mixed Hodge structure on the fundamental group of a punctured curve

Let X be a smooth projective curve of genus $$\ge 1$$ over $${{\mathbb {C}}}$$ , and $$e,\infty \in X({{\mathbb {C}}})$$ be distinct points. Let $$L_n$$ be the mixed Hodge structure of functions on $$\pi _1(X-\{\infty \},e)$$ given by iterated integrals of length $$\le n$$ (as defined by Hain). Building on a work of Darmon et al. (Publications Mathematiques de Besancon 2:19–46, 2012), we express the mixed Hodge extension $${\mathbb {E}}^\infty _{n,e}$$ given by the weight filtration on $$\frac{L_n}{L_{n-2}}$$ as the Abel–Jacobi image of a null-homologous algebraic cycle on $$X^{2n-1}$$ . This algebraic cycle is constructed using the different embeddings of $$X^{n-1}$$ into $$X^n$$ . As a corollary, we show that the extension $${\mathbb {E}}^\infty _{n,e}$$ determines the point $$\infty \in X-\{e\}$$ . When $$n=2$$ , our main result is a strengthening of a theorem of Darmon et al. (Publications Mathematiques de Besancon 2:19–46, 2012). In the final section we assume that $$X,e,\infty $$ are defined over a subfield K of $${{\mathbb {C}}}$$ . Generalizing a construction in (Darmon et al., Publications Mathematiques de Besancon 2:19–46, 2012), we use the extension $${\mathbb {E}}^\infty _{n,e}$$ to define a family of K-rational points on the Jacobian of X parametrized by $$(n-1)$$ -dimensional algebraic cycles on $$X^{2n-2}$$ defined over K.

Picard Group and Fundamental Group of the Moduli of Higgs Bundles on Curves

Abstract Let X be an irreducible smooth projective curve of genus g ≥ 2 over ℂ. Let MG, Higgs δbe a connected reductive affine algebraic group over ℂ. Let Higgs be the moduli space of semistable principal G-Higgs bundles on X of topological type δ∈π1(G). In this article,we compute the fundamental group and Picard group of

On Jacobian group arithmetic for typical divisors on curves

In a previous joint article with F. Abu Salem, we gave efficient algorithms for Jacobian group arithmetic of "typical" divisor classes on C_{3,4} curves, improving on similar results by other authors. At that time, we could only state that a generic divisor was typical, and hence unlikely to be encountered if one implemented these algorithms over a very large finite field. This article pins down an explicit characterization of these typical divisors, for an arbitrary smooth projective curve of genus g >= 1 having at least one rational point. We give general algorithms for Jacobian group arithmetic with these typical divisors, and prove not only that the algorithms are correct if various divisors are typical, but also that the success of our algorithms provides a guarantee that the resulting output is correct and that the resulting input and/or output divisors are also typical. These results apply in particular to our earlier algorithms for C_{3,4} curves. As a byproduct, we obtain a further speedup of approximately 15% on our previous algorithms for C_{3,4} curves.

Schottky via the punctual Hilbert scheme

We show that a smooth projective curve of genus $g$ can be reconstructed from its polarized Jacobian $(X, \Theta)$ as a certain locus in the Hilbert scheme $\mathrm{Hilb}^d(X)$, for $d=3$ and for $d=g+2$, defined by geometric conditions in terms of the polarization $\Theta$. The result is an application of the Gunning-Welters trisecant criterion and the Castelnuovo-Schottky theorem by Pareschi-Popa and Grushevsky, and its scheme theoretic extension by the authors.

A Castelnuovo–Mumford regularity bound for scrolls

Let X⊆Pr be a scroll of codimension e and degree d over a smooth projective curve of genus g. The purpose of this paper is to prove a linear Castelnuovo–Mumford regularity bound that reg(X)≤d−e+1+g(e−1). This bound works over an algebraically closed field of arbitrary characteristic.