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Journal articles on the topic 'Curve genus'

Author: Grafiati

Published: 2 June 2025

Last updated: 31 July 2025

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1

Mourao, Michael. "Extending Elliptic Curve Chabauty to higher genus curves." Manuscripta Mathematica 143, no. 3-4 (2013): 355–77. http://dx.doi.org/10.1007/s00229-013-0621-2.

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2

Cossidente, A., G. Korchmáros, and F. Torres. "Curves of large genus covered by the hermitian curve." Communications in Algebra 28, no. 10 (2000): 4707–28. http://dx.doi.org/10.1080/00927870008827115.

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3

Bröker, Reinier, Everett W. Howe, Kristin E. Lauter, and Peter Stevenhagen. "Genus-2 curves and Jacobians with a given number of points." LMS Journal of Computation and Mathematics 18, no. 1 (2015): 170–97. http://dx.doi.org/10.1112/s1461157014000461.

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AbstractWe study the problem of efficiently constructing a curve $C$ of genus $2$ over a finite field $\mathbb{F}$ for which either the curve $C$ itself or its Jacobian has a prescribed number $N$ of $\mathbb{F}$-rational points.In the case of the Jacobian, we show that any ‘CM-construction’ to produce the required genus-$2$ curves necessarily takes time exponential in the size of its input.On the other hand, we provide an algorithm for producing a genus-$2$ curve with a given number of points that, heuristically, takes polynomial time for most input values. We illustrate the practical applica

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4

Baba, Srinath, and Håkan Granath. "Genus 2 Curves with Quaternionic Multiplication." Canadian Journal of Mathematics 60, no. 4 (2008): 734–57. http://dx.doi.org/10.4153/cjm-2008-033-7.

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AbstractWe explicitly construct the canonical rational models of Shimura curves, both analytically in terms of modular forms and algebraically in terms of coefficients of genus 2 curves, in the cases of quaternion algebras of discriminant 6 and 10. This emulates the classical construction in the elliptic curve case. We also give families of genus 2 QMcurves, whose Jacobians are the corresponding abelian surfaces on the Shimura curve, and with coefficients that are modular forms of weight 12. We apply these results to show that our j-functions are supported exactly at those primes where the gen

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5

Bruin, Nils. "The arithmetic of Prym varieties in genus 3." Compositio Mathematica 144, no. 2 (2008): 317–38. http://dx.doi.org/10.1112/s0010437x07003314.

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AbstractGiven a curve of genus 3 with an unramified double cover, we give an explicit description of the associated Prym variety. We also describe how an unramified double cover of a non-hyperelliptic genus 3 curve can be mapped into the Jacobian of a curve of genus 2 over its field of definition and how this can be used to perform Chabauty- and Brauer–Manin-type calculations for curves of genus 5 with an fixed-point-free involution. As an application, we determine the rational points on a smooth plane quartic and give examples of curves of genus 3 and 5 violating the Hasse principle. The meth

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6

Fan, Jing, Xuejun Fan, Ningning Song, and Long Wang. "Hyperelliptic Covers of Different Degree for Elliptic Curves." Mathematical Problems in Engineering 2022 (July 4, 2022): 1–11. http://dx.doi.org/10.1155/2022/9833393.

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In elliptic curve cryptography (ECC) and hyperelliptic curve cryptography (HECC), the size of cipher-text space defined by the cardinality of Jacobian is a significant factor to measure the security level. Counting problems on Jacobians of elliptic curve can be solved in polynomial time by Schoof–Elkies–Atkin (SEA) algorithm. However, counting problems on Jacobians of hyperelliptic curves are solved less satisfactorily than those on elliptic curves. So, we consider the construction of the cover map from the hyperelliptic curves to the elliptic curves to convert point counting problems on hyper

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7

Moreno-Mejía, Israel. "A Canonical Curve of Genus 17." Results in Mathematics 66, no. 1-2 (2014): 65–86. http://dx.doi.org/10.1007/s00025-014-0364-8.

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8

Balakrishnan, Jennifer S., Netan Dogra, J. Steffen Müller, Jan Tuitman, and Jan Vonk. "Quadratic Chabauty for modular curves: algorithms and examples." Compositio Mathematica 159, no. 6 (2023): 1111–52. http://dx.doi.org/10.1112/s0010437x23007170.

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We describe how the quadratic Chabauty method may be applied to determine the set of rational points on modular curves of genus $g>1$ whose Jacobians have Mordell–Weil rank $g$ . This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or non-trivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin–Lehner quotients $X_0^+(N)$ of pr

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9

Pál, Ambrus. "Solvable Points on Projective Algebraic Curves." Canadian Journal of Mathematics 56, no. 3 (2004): 612–37. http://dx.doi.org/10.4153/cjm-2004-028-0.

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AbstractWe examine the problem of finding rational points defined over solvable extensions on algebraic curves defined over general fields. We construct non-singular, geometrically irreducible projective curves without solvable points of genus g, when g is at least 40, over fields of arbitrary characteristic. We prove that every smooth, geometrically irreducible projective curve of genus 0, 2, 3 or 4 defined over any field has a solvable point. Finally we prove that every genus 1 curve defined over a local field of characteristic zero with residue field of characteristic p has a divisor of deg

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10

Sáez, Meritxell. "Classification of degree two curves in the symmetric square with positive self-intersection." Advances in Geometry 18, no. 2 (2018): 161–80. http://dx.doi.org/10.1515/advgeom-2017-0046.

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Abstract We give a precise classification of the pairs (C, B͠) with C a smooth curve of genus g and B͠ ⊂ C(2) a curve of degree two and positive self-intersection. We prove that there are no such pairs if g < pa(B͠) < 2g−1. We study the singularities and self-intersection of any degree two curve in C(2). Moreover, we give examples of curves with arithmetic genus in the Brill–Noether range and positive self-intersection on C × C.

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11

GUÀRDIA, J. "EXPLICIT GEOMETRY ON A FAMILY OF CURVES OF GENUS 3." Journal of the London Mathematical Society 64, no. 2 (2001): 299–310. http://dx.doi.org/10.1112/s0024610701002538.

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An explicit geometrical study of the curves[formula here]is presented. These are non-singular curves of genus 3, defined over ℚ(a). By exploiting their symmetries, it is possible to determine most of their geometric invariants, such as their bitangent lines and their period lattice. An explicit description is given of the bijection induced by the Abel–Jacobi map between their bitangent lines and odd 2-torsion points on their jacobian. Finally, three elliptic quotients of these curves are constructed that provide a splitting of their jacobians. In the case of the curve [Cscr ]1±√2, which is iso

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12

Knutsen, Andreas Leopold, Margherita Lelli-Chiesa, and Giovanni Mongardi. "Severi varieties and Brill–Noether theory of curves on abelian surfaces." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 749 (2019): 161–200. http://dx.doi.org/10.1515/crelle-2016-0029.

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Abstract Severi varieties and Brill–Noether theory of curves on K3 surfaces are well understood. Yet, quite little is known for curves on abelian surfaces. Given a general abelian surface S with polarization L of type {(1,n)} , we prove nonemptiness and regularity of the Severi variety parametrizing δ-nodal curves in the linear system {|L|} for {0\leq\delta\leq n-1=p-2} (here p is the arithmetic genus of any curve in {|L|} ). We also show that a general genus g curve having as nodal model a hyperplane section of some {(1,n)} -polarized abelian surface admits only finitely many such models up t

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13

Ciliberto, Ciro, and Angelo Felice Lopez. "On the number of moduli of extendable canonical curves." Nagoya Mathematical Journal 167 (2002): 101–15. http://dx.doi.org/10.1017/s0027763000025459.

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AbstractLet C ⊂ ℙg−1 be a canonical curve of genus g. In this article we study the problem of extendability of C, that is when there is a surface S ⊂ ℙg different from a cone and having C as hyperplane section. Using the work of Epema we give a bound on the number of moduli of extendable canonical curves. This for example implies that a family of large dimension of curves that are cover of another curve has general member nonextendable. Using a theorem of Wahl we prove the surjectivity of the Wahl map for the general k-gonal curve of genus g when k = 5, g ≥ 15 or k = 6, g ≥ 13 or k ≥ 7, g ≥ 12

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14

Dülger, Alev. "Multiple curves on punctured orientable surfaces." Filomat 36, no. 20 (2022): 6929–43. http://dx.doi.org/10.2298/fil2220929d.

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We describe each multiple curve on an orientable surface of genus-1, with n punctures and one boundary component by using this multiple curve?s geometric intersection numbers with the embedded curves in this surface.

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15

CHO, KYUNG-HYE, CHANGHO KEEM, and AKIRA OHBUCHI. "ON THE VARIETY OF SPECIAL LINEAR SYSTEMS OF DEGREE g-1 ON SMOOTH ALGEBRAIC CURVES." International Journal of Mathematics 13, no. 01 (2002): 11–29. http://dx.doi.org/10.1142/s0129167x02001204.

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We classify smooth projective algebraic curves C of genus g such that the variety of special linear systems [Formula: see text] has dimension g- 7. We first prove that if [Formula: see text] has dimension g-7≥0 then C is either trigonal, tetragonal, a double covering of a curve of genus 2 or a smooth plane sextic. This result establishes the next extension of dimension theorems of H. Martens and D. Mumford on the variety of special linear systems with the fullest possible generality. We then proceed to show that, under the assumption g≥11, [Formula: see text] has dimension g- 7 if and only if

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16

Lanteri, Antonio, and Hidetoshi Maeda. "Ample Vector Bundles of Curve Genus One." Canadian Mathematical Bulletin 42, no. 2 (1999): 209–13. http://dx.doi.org/10.4153/cmb-1999-025-9.

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AbstractWe investigate the pairs (X, ε) consisting of a smooth complex projective variety X of dimension n and an ample vector bundle ε of rank n − 1 on X such that ε has a section whose zero locus is a smooth elliptic curve.

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17

Fisher, Tom. "The invariants of a genus one curve." Proceedings of the London Mathematical Society 97, no. 3 (2008): 753–82. http://dx.doi.org/10.1112/plms/pdn021.

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18

Fisher, Tom. "The Hessian of a genus one curve." Proceedings of the London Mathematical Society 104, no. 3 (2011): 613–48. http://dx.doi.org/10.1112/plms/pdr039.

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19

Rhoads, James E., J. Richard, III Gott, and Marc Postman. "The genus curve of the Abell clusters." Astrophysical Journal 421 (January 1994): 1. http://dx.doi.org/10.1086/173619.

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20

Korchmáros, Gábor, and Fernando Torres. "On the genus of a maximal curve." Mathematische Annalen 323, no. 3 (2002): 589–608. http://dx.doi.org/10.1007/s002080200316.

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21

Pumplün, Susanne. "Quaternion Algebras over Curves of Genus One Without Rational Points." Algebra Colloquium 12, no. 01 (2005): 67–92. http://dx.doi.org/10.1142/s1005386705000076.

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Quaternion algebras over curves of genus one without rational points over perfect fields of characteristic not two are classified. As a consequence, all quaternion algebras over the function field of such a curve, which are unramified everywhere, are classified.

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22

Bastien, Guy, and Marc Rogalski. "Dynamical systems associated with QRT families of degree four biquadratic curves each of them with genus zero." Sarajevo Journal of Mathematics 16, no. 2 (2022): 187–99. http://dx.doi.org/10.5644/sjm.16.02.05.

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We give some examples of QRT families of biquadratic curves which are of degree 4 (with high degree term $x^2y^2$), but such that every curve of such a family is singular, with genus 0, or is reducible (for some finite set of values of the parameter). This contrasts with classical examples of QRT families studied by many authors. We give also examples of QRT families of degree 4 whose every curve is reducible. Then we sketch out to study the dynamical systems associated to such QRT families of genus zero.

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23

Booker, Andrew R., Jeroen Sijsling, Andrew V. Sutherland, John Voight, and Dan Yasaki. "A database of genus-2 curves over the rational numbers." LMS Journal of Computation and Mathematics 19, A (2016): 235–54. http://dx.doi.org/10.1112/s146115701600019x.

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We describe the construction of a database of genus-$2$curves of small discriminant that includes geometric and arithmetic invariants of each curve, its Jacobian, and the associated$L$-function. This data has been incorporated into the$L$-Functions and Modular Forms Database (LMFDB).

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24

De Cataldo, Mark Andrea A. "The genus of curves on the three dimensional quadric." Nagoya Mathematical Journal 147 (September 1997): 193–211. http://dx.doi.org/10.1017/s0027763000006383.

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AbstractBy means of an ad hoc modification of the so-called “Castelnuovo-Harris analysis” we derive an upper bound for the genus of integral curves on the three dimensional nonsingular quadric which lie on an integral surface of degree 2/c, as a function of k and the degree d of the curve. In order to obtain this we revisit the Uniform Position Principle to make its use computation-free. The curves which achieve this bound can be conveniently characterized.

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25

Kempf, George R. "The Equations Defining a Curve of Genus 4." Proceedings of the American Mathematical Society 97, no. 2 (1986): 219. http://dx.doi.org/10.2307/2046502.

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26

Gupta, Daya, Asok De, and Kakali Chatterjee. "Performance Study of genus 3 Hyperelliptic Curve Cryptosystem." Journal of Information Processing Systems 8, no. 1 (2012): 145–58. http://dx.doi.org/10.3745/jips.2012.8.1.145.

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27

Seidel, Paul. "Homological Mirror Symmetry for the genus two curve." Journal of Algebraic Geometry 20, no. 4 (2011): 727–69. http://dx.doi.org/10.1090/s1056-3911-10-00550-3.

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28

Ma, Jiming. "Hyperbolicity of the genus two separating curve complex." Geometriae Dedicata 152, no. 1 (2010): 147–51. http://dx.doi.org/10.1007/s10711-010-9549-9.

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29

Kempf, George R. "The equations defining a curve of genus $4$." Proceedings of the American Mathematical Society 97, no. 2 (1986): 219. http://dx.doi.org/10.1090/s0002-9939-1986-0835869-2.

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30

Maeda, H., and A. J. Sommese. "Very ample vector bundles of curve genus two." Archiv der Mathematik 79, no. 1 (2002): 74–80. http://dx.doi.org/10.1007/s00013-002-8287-0.

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31

Parlier, Hugo, and Bram Petri. "The Genus of Curve, Pants and Flip Graphs." Discrete & Computational Geometry 59, no. 1 (2017): 1–30. http://dx.doi.org/10.1007/s00454-017-9922-7.

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32

Nguyen, Xuan Tho. "Rational points on a certain genus 2 curve." Comptes Rendus. Mathématique 361, G6 (2023): 1071–73. http://dx.doi.org/10.5802/crmath.471.

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33

Thompson, Abigail. "The disjoint curve property and genus 2 manifolds." Topology and its Applications 97, no. 3 (1999): 273–79. http://dx.doi.org/10.1016/s0166-8641(98)00063-7.

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34

Zhao, Junyan. "Moduli of genus six curves and K-stability." Transactions of the American Mathematical Society, Series B 11, no. 26 (2024): 863–900. http://dx.doi.org/10.1090/btran/195.

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The K-moduli theory provides different compactifications of various moduli spaces, including moduli of curves. As a general genus six curve can be canonically embedded into the smooth quintic del Pezzo surface, we study in this paper the K-moduli spaces M ¯ K ( c ) \overline {M}^K(c) of the quintic log Fano pairs. We classify the strata of genus six curves C C appearing in the K-moduli by explicitly describing the wall-crossing structure. The K-moduli spaces interpolate between two birational moduli spaces constructed by Geometric Invariant Theory (GIT) and moduli of K3 surfaces via Hodge theo

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35

GRZEGORCZYK, I., V. MERCAT, and P. E. NEWSTEAD. "STABLE BUNDLES OF RANK 2 WITH FOUR SECTIONS." International Journal of Mathematics 22, no. 12 (2011): 1743–62. http://dx.doi.org/10.1142/s0129167x11007434.

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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 expre

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36

Kani, Ernst. "Relations Between the Genera and Between the Hasse-Witt Invariants of Galois Coverings of Curves." Canadian Mathematical Bulletin 28, no. 3 (1985): 321–27. http://dx.doi.org/10.4153/cmb-1985-038-0.

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AbstractLet G ⊂ Aut (C) be a (finite) group of automorphisms of a curve C defined over a field K and, for each subgroup H ≤ G, let gH denote the genus of the quotient curve CH = C/H (briefly: quotient genus of H).In this paper we show that certain idempotent relations in the rational group ring [G] imply relations between the quotient genera {gH}H=G this generalizes two theorems of Accola. Moreover, we show that in the case of char (K) = p ≠ 0, a similar statement holds for the Hasse-Witt invariants σH of the curves CH

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37

Lax, R. F. "Weierstrass points on rational nodal curves." Glasgow Mathematical Journal 29, no. 1 (1987): 131–40. http://dx.doi.org/10.1017/s0017089500006741.

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C. Widland [14] has defined Weierstrass points on integral, projective Gorenstein curves. We show here that the Weierstrass points on a generic integral rational nodal curve have the minimal possible weights or, equivalently, that such a curve has the maximum possible number of distinct nonsingular Weierstrass points. Rational curves with g nodes arise in degeneration arguments involving smooth curves of genus g and they have also recently arisen in connection with g-soliton solutions to certain nonlinear partial differential equations [11], [13].

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38

Abu Salem, Fatima K., and Kamal khuri-makdisi. "Fast Jacobian Group Operations for C3,4 Curves over a Large Finite Field." LMS Journal of Computation and Mathematics 10 (2007): 307–28. http://dx.doi.org/10.1112/s146115700000142x.

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Let C be an arbitrary smooth algebraic curve of genus g over a large finite field F. The authors of this paper revisit fast addition algorithms in the Jacobian of C due to Khuri-Makdisi [math.NT/0409209, to appear in Mathematics of Computation]. The algorithms, which reduce to linear algebra in vector spaces of dimension O(g) once |K| ≫ g and which asymptotically require O(g2.376) field operations using fast linear algebra, are shown to perform efficiently even for certain low genus curves. Specifically, the authors provide explicit formulae for performing the group law on Jacobians of C3,4 cu

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39

ARTAL BARTOLO, E., J. I. COGOLLUDO-AGUSTÍN, and A. LIBGOBER. "ALBANESE VARIETIES OF CYCLIC COVERS OF THE PROJECTIVE PLANE AND ORBIFOLD PENCILS." Nagoya Mathematical Journal 227 (October 5, 2016): 189–213. http://dx.doi.org/10.1017/nmj.2016.54.

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The paper studies a relation between fundamental group of the complement to a plane singular curve and the orbifold pencils containing it. The main tool is the use of Albanese varieties of cyclic covers ramified along such curves. Our results give sufficient conditions for a plane singular curve to belong to an orbifold pencil, that is, a pencil of plane curves with multiple fibers inducing a map onto an orbifold curve whose orbifold fundamental group is nontrivial. We construct an example of a cyclic cover of the projective plane which is an abelian surface isomorphic to the Jacobian of a cur

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40

Mochizuki, Shinichi. "Galois Sections in Absolute Anabelian Geometry." Nagoya Mathematical Journal 179 (2005): 17–45. http://dx.doi.org/10.1017/s0027763000025599.

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AbstractWe show that isomorphisms between arithmetic fundamental groups of hyperbolic curves over p-adic local fields preserve the decomposition groups of all closed points (respectively, closed points arising from torsion points of the underlying elliptic curve), whenever the hyperbolic curves in question are isogenous to hyperbolic curves of genus zero defined over a number field (respectively, are once-punctured elliptic curves [which are not necessarily defined over a number field]). We also show that, under certain conditions, such isomorphisms preserve certain canonical “integral structu

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41

Fernández, Julio, Josep González, and Joan-C. Lario. "Plane Quartic Twists of X(5, 3)." Canadian Mathematical Bulletin 50, no. 2 (2007): 196–205. http://dx.doi.org/10.4153/cmb-2007-021-8.

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AbstractGiven an odd surjective Galois representation ϱ: Gℚ → PGL2(3) and a positive integer N, there exists a twisted modular curve X(N, 3)ϱ defined over ℚ whose rational points classify the quadratic ℚ-curves of degree N realizing ϱ. This paper gives a method to provide an explicit plane quartic model for this curve in the genus-three case N = 5.

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42

Brandt, Madeline, and Paul Alexander Helminck. "Tropical superelliptic curves." Advances in Geometry 20, no. 4 (2020): 527–51. http://dx.doi.org/10.1515/advgeom-2020-0014.

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AbstractWe present an algorithm for computing the Berkovich skeleton of a superelliptic curve yn = f(x) over a valued field. After defining superelliptic weighted metric graphs, we show that each one is realizable by an algebraic superelliptic curve when n is prime. Lastly, we study the locus of superelliptic weighted metric graphs inside the moduli space of tropical curves of genus g.

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43

Little, John B. "Distribution of Weierstrass Points on Rational Cuspidal Curves." Canadian Mathematical Bulletin 33, no. 2 (1990): 184–89. http://dx.doi.org/10.4153/cmb-1990-031-7.

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AbstractWe study the set W(𝓛) of Weierstrass points of all positive tensor powers of an invertible sheaf 𝓛 on an irreducible rational curve X with g ≧ 2 ordinary cusps. Using an idea from B. Olsen's study of the analogous question on smooth curves, and an explicit formula for the "theta function" of a cuspidal rational curve, we show that W(𝓛) is never dense on X (in contrast to the case of smooth curves of genus g ≧ 2).

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44

Habegger, Philipp, and Fabien Pazuki. "Bad reduction of genus curves with CM jacobian varieties." Compositio Mathematica 153, no. 12 (2017): 2534–76. http://dx.doi.org/10.1112/s0010437x17007424.

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We show that a genus $2$ curve over a number field whose jacobian has complex multiplication will usually have stable bad reduction at some prime. We prove this by computing the Faltings height of the jacobian in two different ways. First, we use a known case of the Colmez conjecture, due to Colmez and Obus, that is valid when the CM field is an abelian extension of the rationals. It links the height and the logarithmic derivatives of an $L$-function. The second formula involves a decomposition of the height into local terms based on a hyperelliptic model. We use the reduction theory of genus

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45

Bujalance, E., G. Gromadzki, and M. Izquierdo. "On real forms of a complex algebraic curve." Journal of the Australian Mathematical Society 70, no. 1 (2001): 134–42. http://dx.doi.org/10.1017/s1446788700002329.

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AbstractTwo projective nonsingular complex algebraic curves X and Y defined over the field R of real numbers can be isomorphic while their sets X(R) and Y(R) of R-rational points could be even non homeomorphic. This leads to the count of the number of real forms of a complex algebraic curve X, that is, those nonisomorphic real algebraic curves whose complexifications are isomorphic to X. In this paper we compute, as a function of genus, the maximum number of such real forms that a complex algebraic curve admits.

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46

Ônishi, Yoshihiro. "Arithmetical Power Series Expansion of the Sigma Function for a Plane Curve." Proceedings of the Edinburgh Mathematical Society 61, no. 4 (2018): 995–1022. http://dx.doi.org/10.1017/s0013091517000463.

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AbstractThe Weierstrass function σ(u) associated with an elliptic curve can be generalized in a natural way to an entire function associated with a higher genus algebraic curve. This generalized multivariate sigma function has been investigated since the pioneering work of Felix Klein. The present paper shows Hurwitz integrality of the coefficients of the power series expansion around the origin of the higher genus sigma function associated with a certain plane curve, which is called an (n,s)-curve or a plane telescopic curve. For the prime (2), the expansion of the sigma function is not Hurwi

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47

Sahni, Varun. "Analysis of Large Scale Structure using Percolation, Genus and Shape Statistics." Symposium - International Astronomical Union 183 (1999): 210–20. http://dx.doi.org/10.1017/s0074180900132541.

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We probe gravitational clustering in N-body simulations using geometrical descriptors sensitive to ‘connectedness’: the genus curve, percolation and shape statistics. As gravitational clustering advances, the density field in N-body simulations shows an increasingly pronounced departure from Gaussianity reflected in the changing shape of the percolation curve and the changing amplitude and shape of the genus curve. We feel that both genus and percolation curves provide complementary probes of large scale structure topology and could be used to discriminate between models of structure formation

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BRAMBILA-PAZ, L., V. MERCAT, P. E. NEWSTEAD, and F. ONGAY. "NONEMPTINESS OF BRILL–NOETHER LOCI." International Journal of Mathematics 11, no. 06 (2000): 737–60. http://dx.doi.org/10.1142/s0129167x00000350.

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Abstract:

Let X be a non-singular algebraic curve of genus g. We prove that the Brill–Noether locus [Formula: see text] is non-empty if d=nd′+d′′ with 0<d′′< 2n, 1≤s≤g, d′≥(s-1)(s+g)/s, n≤d′′+(n-k)g, (d′′,k) ≠(n, n). These results hold for an arbitrary curve of genus ≥ 2, and allow us to construct a region in the associated "Brill–Noether (μ, λ)-map" of points for which the Brill–Noether loci are non-empty. Even for the generic case, the region so constructed extends beyond that defined by the so-called "Teixidor parallelograms". For hyperelliptic curves, the same methods give more extensive and p

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Nguyen, Duc-Manh. "Translation surfaces and the curve graph in genus two." Algebraic & Geometric Topology 17, no. 4 (2017): 2177–237. http://dx.doi.org/10.2140/agt.2017.17.2177.

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Dembélé, Lassina. "An intriguing hyperelliptic Shimura curve quotient of genus 16." Algebra & Number Theory 14, no. 10 (2020): 2713–42. http://dx.doi.org/10.2140/ant.2020.14.2713.

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