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Journal articles on the topic 'Curves. Geometry, Enumerative'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Caporaso, Lucia. "Enumerative Geometry of Plane Curves." Notices of the American Mathematical Society 67, no. 06 (2020): 1. http://dx.doi.org/10.1090/noti2094.

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2

Ran, Z. "Enumerative geometry of singular plane curves." Inventiones Mathematicae 97, no. 3 (1989): 447–65. http://dx.doi.org/10.1007/bf01388886.

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3

Do, Norman, Musashi A. Koyama, and Daniel V. Mathews. "Counting curves on surfaces." International Journal of Mathematics 28, no. 02 (2017): 1750012. http://dx.doi.org/10.1142/s0129167x17500124.

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We consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology: for a compact surface [Formula: see text], with a finite set of points [Formula: see text] fixed on its boundary, how many configurations of disjoint arcs are there on [Formula: see text] whose boundary is [Formula: see text]? We find that this enumerative problem, counting curves on surfaces, has a rich structure. We show that such curve counts obey an effective recursion, in the general spirit of topological recursion, and exhibit quasi-polynomial behavior. This “elementary curve-counting” i

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4

Len, Yoav, and Dhruv Ranganathan. "Enumerative geometry of elliptic curves on toric surfaces." Israel Journal of Mathematics 226, no. 1 (2018): 351–85. http://dx.doi.org/10.1007/s11856-018-1698-9.

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5

ALUFFI, PAOLO, and CAREL FABER. "PLANE CURVES WITH SMALL LINEAR ORBITS II." International Journal of Mathematics 11, no. 05 (2000): 591–608. http://dx.doi.org/10.1142/s0129167x00000301.

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The "linear orbit" of a plane curve of degree d is its orbit in ℙd(d+3)/2 under the natural action of PGL(3). We classify curves with positive dimensional stabilizer, and we compute the degree of the closure of the linear orbits of curves supported on unions of lines. Together with the results of [3], this encompasses the enumerative geometry of all plane curves with small linear orbit. This information will serve elsewhere as an ingredient in the computation of the degree of the orbit closure of an arbitrary plane curve.

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6

Yu, Tony Yue. "Gromov compactness in non-archimedean analytic geometry." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 741 (2018): 179–210. http://dx.doi.org/10.1515/crelle-2015-0077.

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Abstract Gromov’s compactness theorem for pseudo-holomorphic curves is a foundational result in symplectic geometry. It controls the compactness of the moduli space of pseudo-holomorphic curves with bounded area in a symplectic manifold. In this paper, we prove the analog of Gromov’s compactness theorem in non-archimedean analytic geometry. We work in the framework of Berkovich spaces. First, we introduce a notion of Kähler structure in non-archimedean analytic geometry using metrizations of virtual line bundles. Second, we introduce formal stacks and non-archimedean analytic stacks. Then we c

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7

Chiodo, Alessandro. "Towards an enumerative geometry of the moduli space of twisted curves and rth roots." Compositio Mathematica 144, no. 6 (2008): 1461–96. http://dx.doi.org/10.1112/s0010437x08003709.

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AbstractThe enumerative geometry of rth roots of line bundles is crucial in the theory of r-spin curves and occurs in the calculation of Gromov–Witten invariants of orbifolds. It requires the definition of the suitable compact moduli stack and the generalization of the standard techniques from the theory of moduli of stable curves. In a previous paper, we constructed a compact moduli stack by describing the notion of stability in the context of twisted curves. In this paper, by working with stable twisted curves, we extend Mumford’s formula for the Chern character of the Hodge bundle to the di

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8

KOCK, JOACHIM. "Tangency quantum cohomology and characteristic numbers." Anais da Academia Brasileira de Ciências 73, no. 3 (2001): 319–26. http://dx.doi.org/10.1590/s0001-37652001000300002.

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This work establishes a connection between gravitational quantum cohomology and enumerative geometry of rational curves (in a projective homogeneous variety) subject to conditions of infinitesimal nature like, for example, tangency. The key concept is that of modified psi classes, which are well suited for enumerative purposes and substitute the tautological psi classes of 2D gravity. The main results are two systems of differential equations for the generating function of certain top products of such classes. One is topological recursion while the other is Witten-Dijkgraaf-Verlinde-Verlinde.

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9

RAN, Z. "Bend, break and count II: elliptics, cuspidals, linear genera." Mathematical Proceedings of the Cambridge Philosophical Society 127, no. 1 (1999): 7–12. http://dx.doi.org/10.1017/s030500419900359x.

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In [R2] we showed how elementary considerations involving geometry on ruled surfaces may be used to obtain recursive enumerative formulae for rational plane curves. Here we show how similar considerations may be used to obtain further enumerative formulae, as follow. First some notation. As usual we denote by Ngd the number of irreducible plane curves of degree d and genus g through 3d+g−1 general points. Also, we denote by Ngd→ (resp. Ngd×) the number of such curves passing through general points A1, …, A3d+g−2 and having a given tangent direction (resp. a node) at A1. As is well known and ea

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10

Donovan, Will. "Contractions of 3-folds: Deformations and invariants." International Journal of Mathematics 27, no. 07 (2016): 1640004. http://dx.doi.org/10.1142/s0129167x16400048.

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This paper discusses recent new approaches to studying flopping curves on 3-folds. In a joint paper [Noncommutative deformation and flops, Duke Math. J. 165(8) (2016) 1397–1414], the author and Wemyss introduced a 3-fold invariant, the contraction algebra, which may be associated to such curves. It characterizes their geometric and homological properties in a unified manner, using the theory of noncommutative deformations. Toda has now clarified the enumerative significance of the contraction algebra for flopping curves, calculating its dimension in terms of Gopakumar-Vafa invariants [Noncommu

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11

Bertrand, Benoît, Erwan Brugallé, and Grigory Mikhalkin. "Genus 0 characteristic numbers of the tropical projective plane." Compositio Mathematica 150, no. 1 (2013): 46–104. http://dx.doi.org/10.1112/s0010437x13007409.

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AbstractFinding the so-called characteristic numbers of the complex projective plane$ \mathbb{C} {P}^{2} $is a classical problem of enumerative geometry posed by Zeuthen more than a century ago. For a given$d$and$g$one has to find the number of degree$d$genus$g$curves that pass through a certain generic configuration of points and at the same time are tangent to a certain generic configuration of lines. The total number of points and lines in these two configurations is$3d- 1+ g$so that the answer is a finite integer number. In this paper we translate this classical problem to the correspondin

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12

SHOVAL, MENDY, and EUGENII SHUSTIN. "ON GROMOV–WITTEN INVARIANTS OF DEL PEZZO SURFACES." International Journal of Mathematics 24, no. 07 (2013): 1350054. http://dx.doi.org/10.1142/s0129167x13500547.

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We compute Gromov–Witten invariants of any genus for Del Pezzo surfaces of degree ≥ 2. The genus zero invariants have been computed a long ago [P. Di Francesco and C. Itzykson, Quantum intersection rings, in The Moduli Space of Curves, eds. R. Dijkgraaf et al., Progress in Mathematics, Vol. 129 (Birkhäuser, Boston, 1995), pp. 81–148; L. Göttsche and R. Pandharipande, The quantum cohomology of blow-ups of ℙ2 and enumerative geometry, J. Differential Geom.48(1) (1998) 61–90], Gromov–Witten invariants of any genus for Del Pezzo surfaces of degree ≥ 3 have been found by Vakil [Counting curves on r

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13

Chavarriga, Javier, Jaume Llibre, and Jean Moulin Ollagnier. "On a Result of Darboux." LMS Journal of Computation and Mathematics 4 (2001): 197–210. http://dx.doi.org/10.1112/s1461157000000863.

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AbstractThis paper is concerned with a relation of Darboux in enumerative geometry, which has very useful applications in the study of polynomial vector fields. The original statement of Darboux was not correct. The present paper gives two different elementary proofs of this relation. The first one follows the ideas of Darboux, and uses basic facts about the intersection index of two plane algebraic curves; the second proof is rather more sophisticated, and gives a stronger result, which should also be very useful. The power of the relation of Darboux is then illustrated by the provision of ne

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14

Wall, C. T. C. "Geometry of quartic curves." Mathematical Proceedings of the Cambridge Philosophical Society 117, no. 3 (1995): 415–23. http://dx.doi.org/10.1017/s0305004100073266.

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In recent work [5] which involved enumeration of singularity types of highly singular quintic curves, it was necessary to use rather detailed information on the geometry of quartic curves (for the case when the quintic consists of the quartic and a line). The present paper was written to supply this background. The cases of primary interest for this purpose are the rational quartics, and we concentrate on these.

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15

ZINGER, A. "Enumeration of one-nodal rational curves in projective spaces." Topology 43, no. 4 (2004): 793–829. http://dx.doi.org/10.1016/j.top.2003.10.003.

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16

Basu, Somnath, and Ritwik Mukherjee. "Enumeration of curves with one singular point." Journal of Geometry and Physics 104 (June 2016): 175–203. http://dx.doi.org/10.1016/j.geomphys.2016.02.008.

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17

Zinger, Aleksey. "Enumeration of genus-three plane curves with a fixed complex structure." Journal of Algebraic Geometry 14, no. 1 (2005): 35–81. http://dx.doi.org/10.1090/s1056-3911-04-00375-3.

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18

JINZENJI, MASAO, and MASARU NAGURA. "MIRROR SYMMETRY AND AN EXACT CALCULATION OF AN (N–2)-POINT CORRELATION FUNCTION ON A CALABI-YAU MANIFOLD EMBEDDED IN CPN−1." International Journal of Modern Physics A 11, no. 07 (1996): 1217–52. http://dx.doi.org/10.1142/s0217751x96000559.

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We consider an (N–2)-dimensional Calabi-Yau manifold which is defined as the zero locus of the polynomial of degree N (of the Fermat type) in CPN−1 and its mirror manifold. We introduce an (N–2)-point correlation function (generalized Yukawa coupling) and evaluate it both by solving the Picard-Fuchs equation for period integrals in the mirror manifold and by explicitly calculating the contribution of holomorphic maps of degree 1 to the Yukawa coupling in the Calabi-Yau manifold using the method of algebraic geometry. In enumerating the holomorphic curves in the general-dimensional Calabi-Yau m

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19

Sinha, Rajarishi, Satyandra K. Gupta, Christiaan J. J. Paredis, and Pradeep K. Khosla. "Extracting Articulation Models from CAD Models of Parts With Curved Surfaces." Journal of Mechanical Design 124, no. 1 (2001): 106–14. http://dx.doi.org/10.1115/1.1434267.

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In an assembly, degrees of freedom are realized by creating mating features that permit relative motion between parts. In complex assemblies, interactions between individual degrees of freedom may result in a behavior different from the intended behavior. In addition, current methods perform assembly reasoning by approximating curved surfaces as piecewise linear surfaces. Therefore, it is important to be able to reason about assemblies using exact representations of curved surfaces; verify global motion behavior of parts in the assembly; and create motion simulations of the assembly by examina

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20

Bini, G., I. P. Goulden, and D. M. Jackson. "Transitive Factorizations in the Hyperoctahedral Group." Canadian Journal of Mathematics 60, no. 2 (2008): 297–312. http://dx.doi.org/10.4153/cjm-2008-014-5.

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AbstractThe classical Hurwitz enumeration problem has a presentation in terms of transitive factorizations in the symmetric group. This presentation suggests a generalization from type A to other finite reflection groups and, in particular, to type B. We study this generalization both from a combinatorial and a geometric point of view, with the prospect of providing a means of understanding more of the structure of the moduli spaces of maps with an S2-symmetry. The type A case has been well studied and connects Hurwitz numbers to the moduli space of curves. We conjecture an analogous setting f

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21

Ganor, Yaniv, and Eugenii Shustin. "Enumeration of Unicuspidal Curves of Any Degree and Genus on Toric Surfaces." International Mathematics Research Notices, July 23, 2021. http://dx.doi.org/10.1093/imrn/rnab195.

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Abstract We enumerate complex curves on toric surfaces of any given degree and genus, having a single cusp and nodes as their singularities, and matching appropriately many point constraints. The solution is obtained via tropical enumerative geometry. The same technique applies to enumeration of real plane cuspidal curves: we show that, for any fixed $r\ge 1$ and $d\ge 2r+3$, there exists a generic real $2r$-dimensional linear family of plane curves of degree $d$ in which the number of real $r$-cuspidal curves is asymptotically comparable with the total number of complex $r$-cuspidal curves in

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22

Ran, Ziv. "Enumerative geometry of divisorial families of rational curves." ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, December 14, 2009, 67–85. http://dx.doi.org/10.2422/2036-2145.2004.1.05.

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23

Vakil, R. "The enumerative geometry of rational and elliptic curves in projective space." Journal für die reine und angewandte Mathematik (Crelles Journal) 2000, no. 529 (2000). http://dx.doi.org/10.1515/crll.2000.094.

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24

Huisman, Johannes. "On the enumerative geometry of real algebraic curves having many real branches." Advances in Geometry 3, no. 1 (2003). http://dx.doi.org/10.1515/advg.2003.006.

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25

LOZANO HUERTA, CÉSAR, and TIM RYAN. "ON THE POSITION OF NODES OF PLANE CURVES." Bulletin of the Australian Mathematical Society, June 1, 2020, 1–7. http://dx.doi.org/10.1017/s0004972720000489.

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The Severi variety $V_{d,n}$ of plane curves of a given degree $d$ and exactly $n$ nodes admits a map to the Hilbert scheme $\mathbb{P}^{2[n]}$ of zero-dimensional subschemes of $\mathbb{P}^{2}$ of degree $n$ . This map assigns to every curve $C\in V_{d,n}$ its nodes. For some $n$ , we consider the image under this map of many known divisors of the Severi variety and its partial compactification. We compute the divisor classes of such images in $\text{Pic}(\mathbb{P}^{2[n]})$ and provide enumerative numbers of nodal curves. We also answer directly a question of Diaz–Harris [‘Geometry of the Se

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26

Manzaroli, Matilde. "Real Algebraic Curves on Real del Pezzo Surfaces." International Mathematics Research Notices, July 29, 2020. http://dx.doi.org/10.1093/imrn/rnaa169.

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Abstract The study of the topology of real algebraic varieties dates back to the work of Harnack, Klein, and Hilbert in the 19th century; in particular, the isotopy-type classification of real algebraic curves in real toric surfaces is a classical subject that has undergone considerable evolution. On the other hand, not much is known for more general ambient surfaces. We take a step forward in the study of topological-type classification of real algebraic curves on non-toric surfaces focusing on real del Pezzo surfaces of degree 1 and 2 with multi-components real part. We use degeneration meth

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27

Gillespie, Maria Monks, and Jake Levinson. "Monodromy and K-theory of Schubert curves via generalized jeu de taquin." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings, 28th... (April 22, 2020). http://dx.doi.org/10.46298/dmtcs.6381.

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International audience We establish a combinatorial connection between the real geometry and the K-theory of complex Schubert curves Spλ‚q, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In a previous paper, the second author showed that the real geometry of these curves is described by the orbits of a map ω on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of RP1, with ω as the monodromy operator.We provide a

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28

Goldner, Christoph. "Counting tropical rational space curves with cross-ratio constraints." manuscripta mathematica, June 15, 2021. http://dx.doi.org/10.1007/s00229-021-01317-3.

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AbstractThis is a follow-up paper of Goldner (Math Z 297(1–2):133–174, 2021), where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in Tyomkin (Adv Math 305:1356–1383, 2017) allowed us to use tropical geometry, and, in particular, a degeneration technique called floor diagrams. This correspondence theorem also holds in higher dimension. In the current paper, we introduce so-called cross-ratio floor diagrams and show that they allow us to determine the number of rational space curves that sa

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29

Eremenko, Alexandre, Andrei Gabrielov, Gabriele Mondello, and Dmitri Panov. "Moduli spaces for Lamé functions and Abelian differentials of the second kind." Communications in Contemporary Mathematics, March 26, 2021, 2150028. http://dx.doi.org/10.1142/s0219199721500280.

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The topology of the moduli space for Lamé functions of degree [Formula: see text] is determined: this is a Riemann surface which consists of two connected components when [Formula: see text]; we find the Euler characteristics and genera of these components. As a corollary we prove a conjecture of Maier on degrees of Cohn’s polynomials. These results are obtained with the help of a geometric description of these Riemann surfaces, as quotients of the moduli spaces for certain singular flat triangles. An application is given to the study of metrics of constant positive curvature with one conic si

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30

Eu, Sen-Peng, Tung-Shan Fu, and Yeh-Jong Pan. "The Cyclic Sieving Phenomenon for Faces of Cyclic Polytopes." Electronic Journal of Combinatorics 17, no. 1 (2010). http://dx.doi.org/10.37236/319.

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A cyclic polytope of dimension $d$ with $n$ vertices is a convex polytope combinatorially equivalent to the convex hull of $n$ distinct points on a moment curve in ${\Bbb R}^d$. In this paper, we prove the cyclic sieving phenomenon, introduced by Reiner-Stanton-White, for faces of an even-dimensional cyclic polytope, under a group action that cyclically translates the vertices. For odd-dimensional cyclic polytopes, we enumerate the faces that are invariant under an automorphism that reverses the order of the vertices and an automorphism that interchanges the two end vertices, according to the

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