Projective Algebraic Curve Research Articles

Tensor networks, $p$-adic fields, and algebraic curves: arithmetic and the $\mathrm{AdS}_3 / \mathrm{CFT}_2$ correspondence

One of the many remarkable properties of conformal field theory in two dimensions is its connection to algebraic geometry. Since every compact Riemann surface is a projective algebraic curve, many constructions of interest in physics (which a priori depend on the analytic structure of the spacetime) can be formulated in purely algebraic language. This opens the door to interesting generalizations, obtained by taking another choice of field: for instance, the p-adics. We generalize the AdS/CFT correspondence according to this principle; the result is a formulation of holography in which the bulk geometry is discrete—the Bruhat–Tits tree for PGL(2,Qp)—but the group of bulk isometries nonetheless agrees with that of boundary conformal transformations and is not broken by discretization. We suggest that this forms the natural geometric setting for tensor networks that have been proposed as models of bulk reconstruction via quantum error correcting codes; in certain cases, geodesics in the Bruhat–Tits tree reproduce those constructed using quantum error correction. Other aspects of holography also hold: Standard holographic results for massive free scalar fields in a fixed background carry over to the tree, whose vertical direction can be interpreted as a renormalization-group scale for modes in the boundary CFT. Higher-genus bulk geometries (the BTZ black hole and its generalizations) can be understood straightforwardly in our setting, and the Ryu–Takayanagi formula for the entanglement entropy appears naturally.

Pseudo-real principalG-bundles over a real curve

We consider stable and semistable principal bundles over a smooth projective real algebraic curve, equipped with a real or pseudo-real structure in the sense of Atiyah. After fixing suitable topological invariants, one can build a suitable gauge theory, and show that the resulting moduli spaces of pseudo-real bundles are connected. This in turn allows one to describe the various fixed point varieties on the complex moduli spaces under the action of the real involutions on the curve and the structure group.

On the Maximum Number of Rational Points on Singular Curves over Finite Fields

We give a construction of singular curves with many rational points over finite fields. This construction enables us to prove some results on the maximum number of rational points on an absolutely irreducible projective algebraic curve defined over Fq of geometric genus g and arithmetic genus π.

Moduli of parahoric 𝒢-torsors on a compact Riemann surface

Let X X be an irreducible smooth projective algebraic curve of genus g ≥ 2 g \geq 2 over the ground field C \mathbb {C} , and let G G be a semisimple simply connected algebraic group. The aim of this paper is to introduce the notion of semistable and stable parahoric torsors under a certain Bruhat–Tits group scheme G \mathcal G and to construct the moduli space of semistable parahoric G \mathcal G -torsors; we also identify the underlying topological space of this moduli space with certain spaces of homomorphisms of Fuchsian groups into a maximal compact subgroup of G G . The results give a generalization of the earlier results of Mehta and Seshadri on parabolic vector bundles.

Characterization of the Fermat curve as the most symmetric nonsingular algebraic plane curve

A projective nonsingular plane algebraic curve of degree \(d\ge 4\) is called maximally symmetric if it attains the maximum order of the automorphism groups for complex nonsingular plane algebraic curves of degree \(d\). For \(d\le 7\), all such curves are known. Up to projectivities, they are the Fermat curve for \(d=5,7\); see Kaneta et al. (RIMS Kokyuroku 1109:182–191, 1999) and Kaneta et al. (Geom. Dedic. 85:317–334, 2001), the Klein quartic for \(d=4\), see Hartshorne (Algebraic Geometry. Springer, New York, 1977), and the Wiman sextic for \(d=6\); see Doi et al. (Osaka J. Math. 37:667–687, 2000). In this paper we work on projective plane curves defined over an algebraically closed field of characteristic zero, and we extend this result to every \(d\ge 8\) showing that the Fermat curve is the unique maximally symmetric nonsingular curve of degree \(d\) with \(d\ge 8\), up to projectivity. For \(d=11,13,17,19\), this characterization of the Fermat curve has already been obtained; see Kaneta et al. (Geom. Dedic. 85:317–334, 2001).

Quantum Curves for Hitchin Fibrations and the Eynard–Orantin Theory

We generalize the topological recursion of Eynard–Orantin (JHEP 0612:053, 2006; Commun Number Theory Phys 1:347–452, 2007) to the family of spectral curves of Hitchin fibrations. A spectral curve in the topological recursion, which is defined to be a complex plane curve, is replaced with a generic curve in the cotangent bundle T*C of an arbitrary smooth base curve C. We then prove that these spectral curves are quantizable, using the new formalism. More precisely, we construct the canonical generators of the formal $${\hbar}$$ -deformation family of D modules over an arbitrary projective algebraic curve C of genus greater than 1, from the geometry of a prescribed family of smooth Hitchin spectral curves associated with the $${SL(2,\mathbb{C})}$$ -character variety of the fundamental group π1(C). We show that the semi-classical limit through the WKB approximation of these $${\hbar}$$ -deformed D modules recovers the initial family of Hitchin spectral curves.

Computing Singular Points of Projective Plane Algebraic Curves by Homotopy Continuation Methods

We present an algorithm that computes the singular points of projective plane algebraic curves and determines their multiplicities and characters. The feasibility of the algorithm is analyzed. We prove that the algorithm has the polynomial time complexity on the degree of the algebraic curve. The algorithm involves the combined applications of homotopy continuation methods and a method of root computation of univariate polynomials. Numerical experiments show that our algorithm is feasible and efficient.

The uniformization of certain algebraic hypergeometric functions

The hypergeometric functions Fn−1n are higher transcendental functions, but for certain parameter values they become algebraic, because the monodromy of the defining hypergeometric differential equation becomes finite. It is shown that many algebraic Fn−1nʼs, for which the finite monodromy is irreducible but imprimitive, can be represented as combinations of certain explicitly algebraic functions of a single variable; namely, the roots of trinomials. This generalizes a result of Birkeland, and is derived as a corollary of a family of binomial coefficient identities that is of independent interest. Any tuple of roots of a trinomial traces out a projective algebraic curve, and it is also determined when this so-called Schwarz curve is of genus zero and can be rationally parametrized. Any such parametrization yields a hypergeometric identity that explicitly uniformizes a family of algebraic Fn−1nʼs. Many examples of such uniformizations are worked out explicitly. Even when the governing Schwarz curve is of positive genus, it is shown how it is sometimes possible to construct explicit single-valued or multivalued parametrizations of individual algebraic Fn−1nʼs, by parametrizing a quotiented Schwarz curve. The parametrization requires computations in rings of symmetric polynomials.

Curves, dynamical systems, and weighted point counting

Suppose X is a (smooth projective irreducible algebraic) curve over a finite field k. Counting the number of points on X over all finite field extensions of k will not determine the curve uniquely. Actually, a famous theorem of Tate implies that two such curves over k have the same zeta function (i.e., the same number of points over all extensions of k) if and only if their corresponding Jacobians are isogenous. We remedy this situation by showing that if, instead of just the zeta function, all Dirichlet L-series of the two curves are equal via an isomorphism of their Dirichlet character groups, then the curves are isomorphic up to "Frobenius twists", i.e., up to automorphisms of the ground field. Because L-series count points on a curve in a "weighted" way, we see that weighted point counting determines a curve. In a sense, the result solves the analogue of the isospectrality problem for curves over finite fields (also know as the "arithmetic equivalence problem"): It states that a curve is determined by "spectral" data, namely, eigenvalues of the Frobenius operator of k acting on the cohomology groups of all ℓ-adic sheaves corresponding to Dirichlet characters. The method of proof is to show that this is equivalent to the respective class field theories of the curves being isomorphic as dynamical systems, in a sense that we make precise.

On the Hilbert scheme of the moduli space of vector bundles over an algebraic curve

Let M(n, ξ) be the moduli space of stable vector bundles of rank n ≥ 3 and fixed determinant ξ over a complex smooth projective algebraic curve X of genus g ≥ 4. We use the gonality of the curve and r-Hecke morphisms to describe a smooth open set of an irreducible component of the Hilbert scheme of M(n, ξ), and to compute its dimension. We prove similar results for the scheme of morphisms \({M or_P (\mathbb{G}, M(n, \xi))}\) and the moduli space of stable bundles over \({X \times \mathbb{G}}\), where \({\mathbb{G}}\) is the Grassmannian \({\mathbb{G}(n - r, \mathbb{C}^n)}\). Moreover, we give sufficient conditions for \({M or_{2ns}(\mathbb{P}^1, M(n, \xi))}\) to be non-empty, when s ≥ 1.

Galois actions on regular dessins of small genera

Dessins d'enfants can be regarded as bipartite graphs embedded in compact orientable surfaces. According to Grothendieck and others, a dessin uniquely determines a complex structure on the surface, and even an algebraic structure (as a projective algebraic curve defined over a number field). The general problem of how to determine all properties of the curve from the combinatorics of the dessin is far from being solved. For regular dessins, which are those having an edge-transitive automorphism group, the situation is easier: currently available methods in combinatorial and computational group theory allow the determination of the fields of definition for all curves with regular dessins of genus 2 to 18.

On bundles of rank 3 computing Clifford indices

Let $C$ be a smooth irreducible projective algebraic curve defined over the complex numbers. The notion of the Clifford index of $C$ was extended a few years ago to semistable bundles of any rank. Recent work has been focussed mainly on the rank-2 Clifford index, although interesting results have also been obtained for the case of rank 3. In this paper we extend this work, obtaining improved lower bounds for the rank-3 Clifford index. This allows the first computations of the rank-3 index in non-trivial cases and examples for which the rank-3 index is greater than the rank-2 index.

CERTAIN VALUES OF GAUSSIAN HYPERGEOMETRIC SERIES AND A FAMILY OF ALGEBRAIC CURVES

Let λ ∈ ℚ\{0, -1} and l ≥ 2. Denote by Cl, λ the nonsingular projective algebraic curve over ℚ with affine equation given by [Formula: see text] In this paper, we give a relation between the number of points on Cl, λ over a finite field and Gaussian hypergeometric series. We also give an alternate proof of a result of [D. McCarthy, 3F2 Hypergeometric series and periods of elliptic curves, Int. J. Number Theory6(3) (2010) 461–470]. We find some special values of 3F2 and 2F1 Gaussian hypergeometric series. Finally we evaluate the value of 3F2(4) which extends a result of [K. Ono, Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc.350(3) (1998) 1205–1223].

Hypergeometric functions and a family of algebraic curves

Let λ∈ℚ∖{0,1} and l≥2, and denote by Cl,λ the nonsingular projective algebraic curve over ℚ with affine equation given by $$y^l=x(x-1)(x-\lambda).$$ In this paper, we define Ω(Cl,λ) analogous to the real periods of elliptic curves and find a relation with ordinary hypergeometric series. We also give a relation between the number of points on Cl,λ over a finite field and Gaussian hypergeometric series. Finally, we give an alternate proof of a result of Rouse (Ramanujan J. 12(2):197–205, 2006).

STABLE BUNDLES OF RANK 2 WITH FOUR SECTIONS

This paper contains results on stable bundles of rank 2 with space of sections of dimension 4 on a smooth irreducible projective algebraic curve C. There is a known lower bound on the degree for the existence of such bundles; the main result of the paper is a geometric criterion for this bound to be attained. For a general curve C of genus 10, we show that the bound cannot be attained, but that there exist Petri curves of this genus for which the bound is sharp. We interpret the main results for various curves and in terms of Clifford indices and coherent systems. The results can also be expressed in terms of Koszul cohomology and the methods provide a useful tool for the study of the geometry of the moduli space of curves.

On Weierstraß semigroups at one and two points and their corresponding Poincaré series

The aim of this paper is to introduce and investigate the Poincar\'e series associated with the Weierstra{\ss} semigroup of one and two rational points at a (not necessarily irreducible) non-singular projective algebraic curve defined over a finite field, as well as to describe their functional equations in the case of an affine complete intersection.

FIELD OF MODULI OF GENERALIZED FERMAT CURVES

As a consequence of the Riemann-Roch theorem, a closed Riemann surface $S$ can be described by a non-singular complex projective algebraic curve $C$. A field of definition for $S$ is any subfield $D$ of $\mathbb{C}$ so that we may choose $C$ to be defined by polynomials in $D[x_0, \ldots, x_n]$. The field of moduli of $S$ is ${\mathbb R}$ if and only if $S$ admits an anticonformal automorphism. In the case that the field of moduli of $S$ is ${\mathbb R}$, then $S$ can be defined over the field of moduli if and only if $S$ admits an anticonformal involution. It may happen that the field of moduli is not a field of definition. In this paper, we consider certain class of closed Riemann surfaces, called generalized Fermat curves. These surfaces are the highest Abelian branched cover of certain orbifolds. In this class of Riemann surfaces, we study the problem of deciding when the field of moduli is ${\mathbb R}$ and when, in such a case, it is a field of definition.

Wilson's map operations on regular dessins and cyclotomic fields of definition

Dessins d'enfants can be seen as bipartite graphs embedded in compact orientable surfaces. According to Grothendieck and others, a dessin uniquely determines a complex structure on the surface, even an algebraic structure as a projective algebraic curve defined over a number field. Combinatorial properties of the dessin should therefore determine the equations and also structural properties of the curve, such as the field of moduli or the field of definition. However, apart from a few series of examples, very few general results concerning such correspondences are known. As a step in this direction, we present a graph theoretic characterisation of certain quasiplatonic curves defined over cyclotomic fields, based on Wilson's operations on maps: these leave invariant the graph but change the cyclic ordering of edges around the vertices, therefore they change the embeddings, and hence the dessins, and hence the conformal and algebraic structure of the underlying curves. Under suitable assumptions, satisfied by many series of regular dessins, these changes coincide with the effect of Galois conjugation. This coincidence allows one to draw conclusions about Galois orbits and fields of definition of dessins. The possibilities afforded by these techniques, and their limitations, are illustrated by a new look at some known examples and a study of dessins based on the regular embeddings of complete graphs.

Note sur la détermination algébrique du groupe fondamental pro-résoluble d'une courbe affine

Let X be a smooth projective algebraic curve of genus g minus r ⩾ 1 points defined over an algebraically closed field k of characteristic p ⩾ 0 . The structure of the largest prime to p quotient of the étale fundamental group is well known by transcendental methods: it is isomorphic to the largest prime to p quotient of a free pro-finite group on 2 g + r − 1 generators. We show that, with purely algebraic means, we can prove the corresponding result for the largest pro-solvable quotient of these groups.

The strange duality conjecture for generic curves

Let X X be a smooth connected projective algebraic curve of genus g ≥ 1 g\geq 1 . The strange duality conjecture connects non-abelian theta functions of rank r r and level k k and those of rank k k and level r r on X X (for S U ( r ) SU(r) and U ⁡ ( k ) \operatorname {U}(k) , respectively). In this paper we prove this conjecture for X X generic in the moduli space of curves of genus g g .