Projective Algebraic Curve Research Articles

On actions of Frobenius morphisms for moduli stacks of principal bundles over algebraic curves

We study the various arithmetic and geometric Frobenius morphisms on the moduli stack of principal bundles over a smooth projective algebraic curve and determine explicitly their actions on the l-adic cohomology of the moduli stack in terms of Chern classes.

Stability in the category of smooth mod-p representations of

Abstract Let $p \geq 5$ be a prime number, and let $G = {\mathrm {SL}}_2(\mathbb {Q}_p)$ . Let $\Xi = {\mathrm {Spec}}(Z)$ denote the spectrum of the centre Z of the pro-p Iwahori–Hecke algebra of G with coefficients in a field k of characteristic p. Let $\mathcal {R} \subset \Xi \times \Xi $ denote the support of the pro-p Iwahori ${\mathrm {Ext}}$ -algebra of G, viewed as a $(Z,Z)$ -bimodule. We show that the locally ringed space $\Xi /\mathcal {R}$ is a projective algebraic curve over ${\mathrm {Spec}}(k)$ with two connected components and that each connected component is a chain of projective lines. For each Zariski open subset U of $\Xi /\mathcal {R}$ , we construct a stable localising subcategory $\mathcal {L}_U$ of the category of smooth k-linear representations of G.

Characterization of Lipschitz normally embedded complex curves

The main result of the paper states that a connected complex affine algebraic curve is LNE in Cn if and only if its germ at any singular point is a finite union of non-singular complex curve germs which are pairwise transverse, and its projective closure is in general position with the hyperplane at infinity. To this aim, we completely characterize complex analytic curves of a compact complex manifold which are LNE (regardless of the given Riemannian structure), and therefore we can relate when a projective algebraic curve is LNE in CPn with the property of its general affine traces being LNE in Cn. This allows us to deduce that Lipschitz classification of LNE curves is topological. We describe a complete invariant for the (outer and inner) Lipschitz equivalence of affine LNE curves and show that most such curves cannot be bi-Lipschitz homeomorphic to a plane one.

Elliptic modular invariants of 4-by-4 matrices

Let A be an n-by-n matrix. The ternary form FA(t,x,y)=det⁡(tIn+xℜ(A)+yℑ(A)) characterizes the numerical range of A. For n=4, we assume that the ternary form FA(t,x,y) is irreducible and the complex projective algebraic curve FA(t,x,y)=0 is elliptic. We prove that the number of analytic curves composing the real algebraic curve of FA(t,x,y)=0 is 2. This result is applied to show that the j-invariant of the elliptic curve FA(t,x,y)=0 is greater than or equal to 1.

Viro–Zvonilov Inequalities for Flexible Curves on an Almost Complex Four-Dimensional Manifold

The restrictions on the topology of nonsingular plane projective real algebraic curves of odd degree, obtained by O. Viro and the author in the paper published in the early 90s, are extended to flexible curves lying on an almost complex four-dimensional manifold. Some examples of real algebraic surfaces and real curves on them prove the sharpness of the obtained inequalities. In addition, it is proved that a compact Lie group smooth action can be lifted to a cyclic branched covering space $ \tilde{X} $ over a closed four-dimensional manifold, and a sufficient condition for $ H_1(\tilde{X})=0 $ was found.

RETRACTED ARTICLE: Curves with prescribed symmetry and associated representations of mapping class groups

Let C be a complex smooth projective algebraic curve endowed with an action of a finite group G such that the quotient curve has genus at least 3. We prove that if the G-curve C is very general for these properties, then the natural map from the group algebra QG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {Q}}}G$$\\end{document} to the algebra of Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {Q}}}$$\\end{document}-endomorphisms of its Jacobian is an isomorphism. We use this to obtain (topological) properties regarding certain virtual linear representations of a mapping class group. For example, we show that the connected component of the Zariski closure of such a representation often acts Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {Q}}}$$\\end{document}-irreducibly in a G-isogeny space of H1(C;Q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1(C; {{\\mathbb {Q}}})$$\\end{document} and with image a Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {Q}}}$$\\end{document}-almost simple group.

Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles

Quantum curves were introduced in the physics literature. We develop a mathematical framework for the case associated with Hitchin spectral curves. In this context, a quantum curve is a Rees $\mathcal{D}$-module on a smooth projective algebraic curve, whose semi-classical limit produces the Hitchin spectral curve of a Higgs bundle. We give a method of quantization of Hitchin spectral curves by concretely constructing one-parameter deformation families of opers. We propose a variant of the topological recursion of Eynard--Orantin and Mirzakhani for the context of singular Hitchin spectral curves. We show that a PDE version of topological recursion provides all-order WKB analysis for the Rees $\mathcal{D}$-modules, defined as the quantization of Hitchin spectral curves associated with meromorphic $SL(2,\mathbb{C})$-Higgs bundles. Topological recursion can be considered as a process of quantization of Hitchin spectral curves. We prove that these two quantizations, one via the construction of families of opers, and the other via the PDE recursion of topological type, agree for holomorphic and meromorphic $SL(2,\mathbb{C})$-Higgs bundles. Classical differential equations such as the Airy differential equation provides a typical example. Through these classical examples, we see that quantum curves relate Higgs bundles, opers, a conjecture of Gaiotto, and quantum invariants, such as Gromov--Witten invariants

RANDOM REAL BRANCHED COVERINGS OF THE PROJECTIVE LINE

AbstractIn this paper, we construct a natural probability measure on the space of real branched coverings from a real projective algebraic curve $(X,c_X)$ to the projective line $(\mathbb{C} \mathbb {P}^1,\textit{conj} )$ . We prove that the space of degree d real branched coverings having “many” real branched points (for example, more than $\sqrt {d}^{1+\alpha }$ , for any $\alpha>0$ ) has exponentially small measure. In particular, maximal real branched coverings – that is, real branched coverings such that all the branched points are real – are exponentially rare.

Pole placement for overdetermined 2D systems

We formulate and solve a pole placement problem by state feedback for overdetermined 2D systems modeled by commutative operator vessels. In this setting, the transfer function of the system is given by a meromorphic bundle map between two holomorphic vector bundles of finite rank over the normalization of a projective plane algebraic curve. The obstruction for a solution is given by an existence of a certain meromorphic bundle map on the input bundle. Reducing to the 1D case, this gives a functional obstruction which is equivalent to the classical pole placement theorem. Our result improves on, and gives a new approach to pole placement even in the classical case, and answers a question of Ball and Vinnikov.

On the disposition of cubic and pair of conics in a real projective plane

In the first part of the 16th Hilbert problem the question about the topology of nonsingular projective algebraic curves and surfaces was formulated. The problem on topology of algebraic manifolds with singularities belong to this subject too. The particular case of this problem is the study of curves that are decompozable into the product of curves in a general position. This paper deals with the problem of topological classification of mutual positions of a nonsingular curve of degree three and two nonsingular curves of degree two in the real projective plane. Additiolal conditions for this problem include general position of the curves and its maximality; in particular, the number of common points for each pair of curves-factors reaches its maximum. It is proved that the classification contains no more than six specific types of positions of the species under study. Four position types are built, and the question of realizability of the two remaining ones is open.

Orthogonal to Principal Ideles

We describe the orthogonal to the group of principal ideles with respect to the global tame symbol pairing on the group of ideles of a smooth projective algebraic curve over a field.

Separable Degree of the Gauss Map and Strict Dual Curves Over Finite Fields

Let $$\mathcal X$$ be a non-degenerate projective algebraic curve and denote by $$\mathcal X^{'}$$ its strict dual curve. The map $$\gamma :\mathcal X\longrightarrow \mathcal X^{'}$$ is called (strict) Gauss map of $$\mathcal X$$ . In this manuscript, we study the separable degree of the Gauss map of curves defined over finite fields. In particular, we give a generalization of a known result on the separable degree of the Gauss map of plane Frobenius nonclassical curves. We also obtain a characterization of certain plane strange curves.

Deligne pairings and families of rank one local systems on algebraic curves

For smooth families $\mathcal{X} \to S$ of projective algebraic curves and holomorphic line bundles $\mathcal{L, M} \to X$ equipped with flat relative connections, we prove the existence of a canonical and functorial “intersection” connection on the Deligne pairing $\langle \mathcal{L, M} \rangle \to S$. This generalizes the construction of Deligne in the case of Chern connections of hermitian structures on $\mathcal{L}$ and $\mathcal{M}$. A relationship is found with the holomorphic extension of analytic torsion, and in the case of trivial fibrations we show that the Deligne isomorphism is flat with respect to the connections we construct. Finally, we give an application to the construction of a meromorphic connection on the hyperholomorphic line bundle over the twistor space of rank one flat connections on a Riemann surface.

Bounds for the number of points on curves over finite fields

Let $$\mathcal{X}$$ be a projective irreducible nonsingular algebraic curve defined over a finite field $$\mathbb{F}_q$$ . This paper presents a variation of the Stohr–Voloch theory and sets new bounds to the number of $${F_{{q^r}}}$$ -rational points on X. In certain cases, where comparison is possible, the results are shown to improve other bounds such as Weil’s, Stohr–Voloch’s and Ihara’s.

Upper bounds for some Brill–Noether loci over a finite field

Let [Formula: see text] be a smooth projective algebraic curve of genus [Formula: see text], over the finite field [Formula: see text]. A classical result of H. Martens states that the Brill–Noether locus of line bundles [Formula: see text] in [Formula: see text] with [Formula: see text] and [Formula: see text] is of dimension at most [Formula: see text], under conditions that hold when such an [Formula: see text] is both effective and special. We show that the number of such [Formula: see text] that are rational over [Formula: see text] is bounded above by [Formula: see text], with an explicit constant [Formula: see text] that grows exponentially with [Formula: see text]. Our proof uses the Weil estimates for function fields, and is independent of Martens’ theorem. We apply this bound to give a precise lower bound of the form [Formula: see text] for the probability that a line bundle in [Formula: see text] is base point free. This gives an effective version over finite fields of the usual statement that a general line bundle of degree [Formula: see text] is base point free. This is applicable to the author’s work on fast Jacobian group arithmetic for typical divisors on curves.

Counting points on the Fricke–Macbeath curve over finite fields

The Fricke-Macbeath curve is a smooth projective algebraic curve of genus 7 with automorphism group PSL₂(픽₈). We recall two models of it (introduced, respectively, by Maxim Hendriks and by Bradley Brock) defined over ℚ, and we establish an explicit isomorphism defined over ℚ( −7 ) between these models. Moreover, we decompose up to isogeny over ℚ the jacobian of one of these models. As a consequence we obtain a simple formula for the number of points over 픽q on (the reduction of) this model, in terms of the elliptic curve with equation y² = x³ + x² − 114x − 127. Moreover, twists by elements of PSL₂(픽₈) of the curve over finite fields are described. The curve leads to a number of new records as maintained on manYPoints of curves of genus 7 with many rational points over finite fields.

A refined notion of arithmetically equivalent number fields, and curves with isomorphic Jacobians

We construct examples of number fields which are not isomorphic but for which their adele groups, the idele groups, and the idele class groups are isomorphic. We also construct examples of projective algebraic curves which are not isomorphic but for which their Jacobian varieties are isomorphic. Both are constructed using an example in group theory provided by Leonard Scott of a finite group G and subgroups H1 and H2 which are not conjugate in G but for which the G-module Z[G/H1] is isomorphic to Z[G/H2].

Remarks on the Geometry of Secant Loci

For a very ample line bundle L on a smooth projective algebraic curve C, we prove that, under some circumstances on positive integers \(\gamma , d\), if one had \(\dim V^{d-1}_{d}(L)=d-1-\gamma \), then the scheme \(V^{\gamma +2}_{\gamma +3}(L)\), if non-empty, would be 2-dimensional. Furthermore, for a 2-very ample line bundle L on C and an integer d, \(4\le d\le h^0(L)-2\), it will be proved that the dimension of the scheme \( V^{d-1}_{d}(L)\) cannot attain its maximum value, i.e. \(d-2\). This will be used to prove irreduciblity of the highest secant loci of L, namely \(V^{h^0(L)-2}_{h^0(L)-1}(L)\). Then we discuss on the existence of very ample line bundles having reducible highest secant loci on k-gonal curves. Extending a well-known result of Montserrat Teixidor to secant loci, our results answer a question proposed and left unanswered recently by Marian Aprodu and Edoardo Sernesi.

Automorphisms of generalized Fermat curves

Let K be an algebraically closed field of characteristic p≥0. A generalized Fermat curve of type (k,n), where k,n≥2 are integers (for p≠0 we also assume that k is relatively prime to p), is a non-singular irreducible projective algebraic curve Fk,n defined over K admitting a group of automorphisms H≅Zkn so that Fk,n/H is the projective line with exactly (n+1) cone points, each one of order k. Such a group H is called a generalized Fermat group of type (k,n). If (n−1)(k−1)>2, then Fk,n has genus gn,k>1 and it is known to be non-hyperelliptic. In this paper, we prove that every generalized Fermat curve of type (k,n) has a unique generalized Fermat group of type (k,n) if (k−1)(n−1)>2 (for p>0 we also assume that k−1 is not a power of p).Generalized Fermat curves of type (k,n) can be described as a suitable fiber product of (n−1) classical Fermat curves of degree k. We prove that, for (k−1)(n−1)>2 (for p>0 we also assume that k−1 is not a power of p), each automorphism of such a fiber product curve can be extended to an automorphism of the ambient projective space. In the case that p>0 and k−1 is a power of p, we use tools from the theory of complete projective intersections in order to prove that, for k and n+1 relatively prime, every automorphism of the fiber product curve can also be extended to an automorphism of the ambient projective space.In this article we also prove that the set of fixed points of the non-trivial elements of the generalized Fermat group coincide with the hyper-osculating points of the fiber product model under the assumption that the characteristic p is either zero or p>kn−1.

Integrable Systems and Torelli Theorems for the Moduli Spaces of Parabolic Bundles and Parabolic Higgs Bundles

AbstractWe prove a Torelli theorem for the moduli space of semistable parabolic Higgs bundles over a smooth complex projective algebraic curve under the assumption that the parabolic weight systemis generic. When the genus is at least two, using this result we also prove a Torelli theoremfor the moduli space of semistable parabolic bundles of rank at least two with generic parabolic weights. The key input in the proofs is a method of J.C. Hurtubise.