On nondegeneracy of curves

Castryck, Wouter; Voight, John

doi:10.2140/ant.2009.3.255

Cited by 31 publications

(64 citation statements)

References 23 publications

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“…The number 2g + 1 in Equation 4 is the dimension of the classical moduli space of trigonal curves of genus g, whose tropicalization is related to our stacky fan M planar g . Our primary source for the relevant material from algebraic geometry is the article [10] by Castryck and Voight. Our paper can be seen as a refined combinatorial extension of theirs.…”

Section: (4)mentioning

confidence: 99%

“…We follow the approach of Castryck and Voight [10] in constructing polygons P that suffice for the union (Equation 3). We write P int for the convex hull of the g interior lattice points of P. This is the interior hull of P. The relationship between the polygons P and P int is studied in polyhedral adjunction theory [14].…”

Section: Proposition 22 the Cone M Is The Image Of The Secondary Comentioning

confidence: 99%

“…That article was a primary source of inspiration for our study. In particular, [10], Theorem 2.1 determined the dimensions of the spaces M planar g for all g. Whenever we speak about the 'dimension expected from classical algebraic geometry' , as we do in Theorem 1.1, this refers to the formulas for dim(M P ) and dim M We summarize the objects discussed so far in a diagram of surjections and inclusions:…”

Section: Algebraic Geometrymentioning

confidence: 99%

“…These polygons appear in [10], Section 12, where they are used to argue that T g defines one of the irreducible components of M planar g , namely, M P . The same P appear in the next section, where they serve to prove one inequality on the dimension in Theorem 1.1.…”

Section: Algebraic Geometrymentioning

confidence: 99%

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Moduli of tropical plane curves

Brodsky

Joswig

Morrison

et al. 2015

Mathematical Sciences

107

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We study the moduli space of metric graphs that arise from tropical plane curves. There are far fewer such graphs than tropicalizations of classical plane curves. For fixed genus g, our moduli space is a stacky fan whose cones are indexed by regular unimodular triangulations of Newton polygons with g interior lattice points. It has dimension 2g + 1 unless g ≤ 3 or g = 7. We compute these spaces explicitly for g ≤ 5.

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Section: (4)mentioning

confidence: 99%

Section: Proposition 22 the Cone M Is The Image Of The Secondary Comentioning

confidence: 99%

Section: Algebraic Geometrymentioning

confidence: 99%

Section: Algebraic Geometrymentioning

confidence: 99%

See 2 more Smart Citations

Moduli of tropical plane curves

Brodsky

Joswig

Morrison

et al. 2015

Mathematical Sciences

107

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show abstract

“…V = a ℓ D ℓ for certain integers a ℓ (where the D ℓ 's are the torus-invariant prime divisors of Tor(Σ ′ )). From our bounds (11) and (12) we see that 3 4 γ 2 ≤ (γ + s) 2 s < (γ + γ) 2 s = 4 s γ 2 which implies that s ≤ 5. Rewrite the first inequality as (3s − 4)γ 2 − 8sγ − 4s 2 ≤ 0: if s ≥ 2 then the largest real root of the left-hand side, when viewed as a polynomial in γ, is given by (4s + 2s √ 3s)/(3s − 4) which for s ≤ 5 is strictly less than 9.…”

Section: Gonalitymentioning

confidence: 69%

Linear Pencils Encoded in the Newton Polygon

Castryck

Cools

2016

Int Math Res Notices

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Abstract. Let C be an algebraic curve defined by a sufficiently generic bivariate Laurent polynomial with given Newton polygon ∆. It is classical that the geometric genus of C equals the number of lattice points in the interior of ∆. In this paper we give similar combinatorial interpretations for the gonality, the Clifford index and the Clifford dimension, by removing a technical assumption from a recent result of Kawaguchi. More generally, the method shows that apart from certain well-understood exceptions, every base-point free pencil whose degree equals or slightly exceeds the gonality is combinatorial, in the sense that it corresponds to projecting C along a lattice direction. Along the way we prove various features of combinatorial pencils. For instance, we give an interpretation for the scrollar invariants associated to a combinatorial pencil, and show how one can tell whether the pencil is complete or not.Among the applications, we find that every smooth projective curve admits at most one Weierstrass semi-group of embedding dimension 2, and that if a non-hyperelliptic smooth projective curve C of genus g ≥ 2 can be embedded in the n th Hirzebruch surface H n , then n is actually an invariant of C.

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Tropical Geometry

Morrison

2020

Foundations for Undergraduate Research in Mathematics

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Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build with these new operations. These equations define piecewise-linear geometric objects called tropical varieties. We explore these tropical varieties in two and three dimensions, building up discrete tools for studying them and determining their geometric properties. We then discuss the relationship between tropical geometry and algebraic geometry, which considers shapes defined by usual polynomial equations.Suggested prerequisites. We use standard set theory notation (unions, functions, etc.) throughout this chapter. Section 1 draws on terminology and motivation from abstract algebra and linear algebra, but can be understood without them. Section 2 draws on topics from discrete geometry, although it is mostly self-contained. Section 3 includes geometry in three dimensions, which uses some notation from a standard course in multivariable calculus. Section 4 uses ring theory terminology from an abstract algebra course.

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On nondegeneracy of curves

Cited by 31 publications

References 23 publications

Moduli of tropical plane curves

Moduli of tropical plane curves

Linear Pencils Encoded in the Newton Polygon

Tropical Geometry

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