scholarly journals Genus 0 characteristic numbers of the tropical projective plane

2013 ◽  
Vol 150 (1) ◽  
pp. 46-104 ◽  
Author(s):  
Benoît Bertrand ◽  
Erwan Brugallé ◽  
Grigory Mikhalkin
Keyword(s):  
Projective Plane ◽  
Tropical Geometry ◽  
Classical Problem ◽  
Integer Number ◽  
Hurwitz Numbers ◽  
Formula One ◽  
Characteristic Numbers ◽  
Pass Through ◽  
Finite Integer ◽  
Correspondence Theorem

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 corresponding enumerative problem of tropical geometry in the case when$g= 0$. Namely, we show that the tropical problem is well posed and establish a special case of the correspondence theorem that ensures that the corresponding tropical and classical numbers coincide. Then we use the floor diagram calculus to reduce the problem to pure combinatorics. As a consequence, we express genus 0 characteristic numbers of$ \mathbb{C} {P}^{2} $in terms of open Hurwitz numbers.

The Study on the Linking-Up of Free-Form Surface Fiber Placement Track Based on Mesh Creating

2008 ◽  
Vol 392-394 ◽  
pp. 682-687 ◽  
Author(s):  
Zhong Xi Shao ◽  
Hong Ya Fu ◽  
De Cai Li
Keyword(s):  
Projective Plane ◽  
Calculation Method ◽  
New Method ◽  
Free Form ◽  
Parallel Projection ◽  
Reference Vector ◽  
Fiber Placement ◽  
Vertex Point ◽  
Free Form Surface ◽  
Pass Through

When using meshing creating method of FP (fiber placement) track, once the track point falls at some vertex point of mesh element, in the meantime the vertex point happens to be shared by several mesh elements, there needs a reasonable calculation method to select next mesh element which the FP track will pass through. Then it comes to the problem on linking of FP tracks. In order to solve it, in this paper, the author puts forward a new method, in which parallel projection theory is used, project need analytical mesh element and FP reference vector to a sound projective plane, on which the mesh element be selected and the FP track be calculated, then the FP track would be projected back to the placement surface. Program using this method realized a reasonable joint at the shared vertex point of meshing elements, which the FP direction has little change, and the mutation of track doesn’t come forth. So, the correctness of the method, which putted forward in this paper, is proved.

scholarly journals Refined curve counting with tropical geometry

2015 ◽  
Vol 152 (1) ◽  
pp. 115-151 ◽  
Author(s):  
Florian Block ◽  
Lothar Göttsche
Keyword(s):  
Tropical Geometry ◽  
Ruled Surfaces ◽  
Plane Curves ◽  
Toric Surfaces ◽  
Formula One ◽  
Tropical Curves ◽  
Tropical Curve ◽  
Severi Variety ◽  
Floor Diagrams

The Severi degree is the degree of the Severi variety parametrizing plane curves of degree $d$ with ${\it\delta}$ nodes. Recently, Göttsche and Shende gave two refinements of Severi degrees, polynomials in a variable $y$, which are conjecturally equal, for large $d$. At $y=1$, one of the refinements, the relative Severi degree, specializes to the (non-relative) Severi degree. We give a tropical description of the refined Severi degrees, in terms of a refined tropical curve count for all toric surfaces. We also refine the equivalent count of floor diagrams for Hirzebruch and rational ruled surfaces. Our description implies that, for fixed ${\it\delta}$, the refined Severi degrees are polynomials in $d$ and $y$, for large $d$. As a consequence, we show that, for ${\it\delta}\leqslant 10$ and all $d\geqslant {\it\delta}/2+1$, both refinements of Göttsche and Shende agree and equal our refined counts of tropical curves and floor diagrams.

scholarly journals Correspondence theorems via tropicalizations of moduli spaces

2016 ◽  
Vol 18 (03) ◽  
pp. 1550043 ◽  
Author(s):  
Andreas Gross
Keyword(s):  
Moduli Spaces ◽  
Toric Variety ◽  
Algebraic Curves ◽  
Rational Curves ◽  
Tropical Curves ◽  
Algebraic Tori ◽  
Pass Through ◽  
General Correspondence ◽  
Correspondence Theorem ◽  
Generic Points

We show that the moduli spaces of irreducible labeled parametrized marked rational curves in toric varieties can be embedded into algebraic tori such that their tropicalizations are the analogous tropical moduli spaces. These embeddings are shown to respect the evaluation morphisms in the sense that evaluation commutes with tropicalization. With this particular setting in mind, we prove a general correspondence theorem for enumerative problems which are defined via “evaluation maps” in both the algebraic and tropical world. Applying this to our motivational example, we show that the tropicalizations of the curves in a given toric variety which intersect the boundary divisors in their interior and with prescribed multiplicities, and pass through an appropriate number of generic points are precisely the tropical curves in the corresponding tropical toric variety satisfying the analogous condition. Moreover, the intersection-theoretically defined multiplicities of the tropical curves are equal to the numbers of algebraic curves tropicalizing to them.

Configurations defined by six lines in space of three dimensions

1933 ◽  
Vol 29 (1) ◽  
pp. 52-68 ◽  
Author(s):  
J. A. Todd
Keyword(s):  
Linear System ◽  
Classical Problem ◽  
Three Dimensions ◽  
Quartic Surface ◽  
Lines In Space ◽  
Pass Through ◽  
Quartic Surfaces

The investigations which follow were originally suggested by the now classical problem of Cayley, the determination of the condition that seven lines in space, of which no two intersect, should lie on a quartic surface. This problem suggests the consideration of the linear system of quartic surfaces which pass through six given lines, and this, essentially, is the basis of all that follows.

scholarly journals Tropical mirror symmetry for elliptic curves

2017 ◽  
Vol 2017 (732) ◽  
pp. 211-246 ◽  
Author(s):  
Janko Böhm ◽  
Kathrin Bringmann ◽  
Arne Buchholz ◽  
Hannah Markwig
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Mirror Symmetry ◽  
Alternative Proof ◽  
Hurwitz Numbers ◽  
Feynman Integrals ◽  
Feynman Graphs ◽  
Correspondence Theorem ◽  
Theoretical Results

Abstract Mirror symmetry relates Gromov–Witten invariants of an elliptic curve with certain integrals over Feynman graphs [10]. We prove a tropical generalization of mirror symmetry for elliptic curves, i.e., a statement relating certain labeled Gromov–Witten invariants of a tropical elliptic curve to more refined Feynman integrals. This result easily implies the tropical analogue of the mirror symmetry statement mentioned above and, using the necessary Correspondence Theorem, also the mirror symmetry statement itself. In this way, our tropical generalization leads to an alternative proof of mirror symmetry for elliptic curves. We believe that our approach via tropical mirror symmetry naturally carries the potential of being generalized to more adventurous situations of mirror symmetry. Moreover, our tropical approach has the advantage that all involved invariants are easy to compute. Furthermore, we can use the techniques for computing Feynman integrals to prove that they are quasimodular forms. Also, as a side product, we can give a combinatorial characterization of Feynman graphs for which the corresponding integrals are zero. More generally, the tropical mirror symmetry theorem gives a natural interpretation of the A-model side (i.e., the generating function of Gromov–Witten invariants) in terms of a sum over Feynman graphs. Hence our quasimodularity result becomes meaningful on the A-model side as well. Our theoretical results are complemented by a Singular package including several procedures that can be used to compute Hurwitz numbers of the elliptic curve as integrals over Feynman graphs.

scholarly journals Enumeration of complex and real surfaces via tropical geometry

2018 ◽  
Vol 18 (1) ◽  
pp. 69-100
Author(s):  
Hannah Markwig ◽  
Thomas Markwig ◽  
Eugenii Shustin
Keyword(s):  
Tropical Geometry ◽  
Three Dimensional ◽  
Algebraic Surfaces ◽  
Lattice Path ◽  
Singular Surfaces ◽  
Dimensional Version ◽  
Correspondence Theorem

AbstractWe prove a correspondence theorem for singular tropical surfaces in ℝ3, which recovers singular algebraic surfaces in an appropriate toric three-fold that tropicalize to a given singular tropical surface. Furthermore, we develop a three-dimensional version of Mikhalkin’s lattice path algorithm that enumerates singular tropical surfaces passing through an appropriate configuration of points in ℝ3. As application we show that there are pencils of real surfaces of degreedin ℙ3containing at least (3/2)d3+O(d2) singular surfaces, which is asymptotically comparable to the number 4(d− 1)3of all complex singular surfaces in the pencil. Our result relies on the classification of singular tropical surfaces [12].

scholarly journals Counting curves on Hirzebruch surfaces: tropical geometry and the Fock space

2021 ◽  
pp. 1-41
Author(s):  
RENZO CAVALIERI ◽  
PAUL JOHNSON ◽  
HANNAH MARKWIG ◽  
DHRUV RANGANATHAN
Keyword(s):  
Fock Space ◽  
Tropical Geometry ◽  
Geometric Interpretation ◽  
Toric Surfaces ◽  
Tropical Curves ◽  
Hirzebruch Surfaces ◽  
Witten Theory ◽  
Toric Degenerations ◽  
Correspondence Theorem ◽  
Floor Diagrams

We study the stationary descendant Gromov–Witten theory of toric surfaces by combining and extending a range of techniques – tropical curves, floor diagrams and Fock spaces. A correspondence theorem is established between tropical curves and descendant invariants on toric surfaces using maximal toric degenerations. An intermediate degeneration is then shown to give rise to floor diagrams, giving a geometric interpretation of this well-known bookkeeping tool in tropical geometry. In the process, we extend floor diagram techniques to include descendants in arbitrary genus. These floor diagrams are then used to connect tropical curve counting to the algebra of operators on the bosonic Fock space, and are showno coincide with the Feynman diagrams of appropriate operators. This extends work of a number of researchers, including Block–Göttsche, Cooper–Pandharipande and Block–Gathmann–Markwig.

scholarly journals Counting tropical rational space curves with cross-ratio constraints

2021 ◽  
Author(s):  
Christoph Goldner
Keyword(s):  
Tropical Geometry ◽  
Rational Curves ◽  
Cross Ratio ◽  
Current Paper ◽  
Higher Dimension ◽  
Rational Space ◽  
Space Curves ◽  
Correspondence Theorem ◽  
Floor Diagrams

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 satisfy general positioned point and cross-ratio conditions. The multiplicities of such cross-ratio floor diagrams can be calculated by enumerating certain rational tropical curves in the plane.

Hirzebruch-type inequalities and plane curve configurations

2017 ◽  
Vol 28 (02) ◽  
pp. 1750013 ◽  
Author(s):  
Piotr Pokora
Keyword(s):  
Projective Plane ◽  
Type Inequality ◽  
Plane Curve ◽  
Complex Projective Plane ◽  
Characteristic Numbers ◽  
Complex Projective

In this paper, we come back to a problem proposed by F. Hirzebruch in the 1980s, namely whether there exists a configuration of smooth conics in the complex projective plane such that the associated desingularization of the Kummer extension is a ball quotient. We extend our considerations to the so-called [Formula: see text]-configurations of curves in the projective plane and we show that in most cases for a given configuration the associated desingularization of the Kummer extension is not a ball quotient. Moreover, we provide improved versions of Hirzebruch-type inequality for [Formula: see text]-configurations. Finally, we show that the so-called characteristic numbers (or [Formula: see text] numbers) for [Formula: see text]-configurations are bounded from above by [Formula: see text]. At the end of the paper we give some examples of surfaces constructed via Kummer extensions branched along conic configurations.

Sign in / Sign up

Export Citation Format

Share Document