The lattice size of a lattice polygon

Castryck, Wouter; Cools, Filip

doi:10.1016/j.jcta.2015.06.005

Cited by 10 publications

(14 citation statements)

References 21 publications

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“…From theénoncé it follows that if ∆ (1) is non-equivalent to any of the polygons listed in (9), then the gonality of U(f ) equals the lattice width of ∆. Thus by Lemma 5.2.…”

Section: Gonalitymentioning

confidence: 87%

“…3.10-3.12,4.3]. We stress that for these bounds it is essential that ∆ max is maximal and that ∆ (1) is not among the polygons listed in (9).…”

Section: Gonalitymentioning

confidence: 99%

“…The method of the previous section can be adapted to show that, apart from some reasonably well-understood exceptional instances of ∆, every near-gonal pencil on a ∆-non-degenerate curve is combinatorial. It is convenient to state our main result in terms of the lattice size, a notion to which we have devoted a separate paper [9]. If ∆ = ∅, then its lattice size is defined as the minimal integer d ≥ 0 such that ∆ is equivalent to a lattice polygon that is contained in dΣ.…”

Section: Near-gonal Pencilsmentioning

confidence: 99%

“…Lemma 5.2. (i)), there exists an expression for ls(∆) in terms of ls(∆ (1) ), allowing one to compute ls(∆) by gradually peeling off the polygon [9,Thm. 3.5].…”

Section: Near-gonal Pencilsmentioning

confidence: 99%

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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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“…From theénoncé it follows that if ∆ (1) is non-equivalent to any of the polygons listed in (9), then the gonality of U(f ) equals the lattice width of ∆. Thus by Lemma 5.2.…”

Section: Gonalitymentioning

confidence: 87%

“…3.10-3.12,4.3]. We stress that for these bounds it is essential that ∆ max is maximal and that ∆ (1) is not among the polygons listed in (9).…”

Section: Gonalitymentioning

confidence: 99%

Section: Near-gonal Pencilsmentioning

confidence: 99%

“…Lemma 5.2. (i)), there exists an expression for ls(∆) in terms of ls(∆ (1) ), allowing one to compute ls(∆) by gradually peeling off the polygon [9,Thm. 3.5].…”

Section: Near-gonal Pencilsmentioning

confidence: 99%

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Linear Pencils Encoded in the Newton Polygon

Castryck

Cools

2016

Int Math Res Notices

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“…This paper is devoted to providing sharp lower bounds on the area of plane convex bodies P , which involve the lattice size of P . This invariant was introduced and studied by Castryck and Cools in [6], where they provided an algorithm for computing the lattice size. The algorithm is based on the idea of Schicho, who introduced in [13] a procedure for mapping a lattice polygon inside a small multiple of the standard simplex.…”

Section: Introductionmentioning

confidence: 99%

Lattice Size and Area of Plane Convex Bodies

Soprunova¹

2021

Preprint

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The lattice size of a lattice polygon P was introduced and studied by Schicho, and by Castryck and Cools in relation to the problem of bounding the total degree and the bi-degree of the defining equation of an algebraic curve. In this paper we establish sharp lower bounds on the area of plane convex bodies P ⊂ R 2 that involve the lattice size of P . In particular, we improve bounds established by Arnold, and Bárány and Pach. We also provide a classification of minimal lattice polygons P ⊂ R 2 of fixed lattice size ls (P ).

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