A Counterexample to Fulton's Conjecture on [formula]0,n

Vermeire, Peter

doi:10.1006/jabr.2001.9044

Cited by 37 publications

(35 citation statements)

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“…It is known that this is not the cone spanned by the boundary divisors. This was observed independently by Keel and Vermeire, [30]. Part of the problem is that given a rational polytope P it is very hard to compute the dual polytope Q.…”

Section: Examples From Moduli Spacesmentioning

confidence: 93%

Mori dream spaces

McKernan

2010

Jpn. J. Math.

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Section: Examples From Moduli Spacesmentioning

confidence: 93%

Mori dream spaces

McKernan

2010

Jpn. J. Math.

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“…We recall a description ofĪ 6 due to Vermeire [12] that uses Kapranov's description [10] ofM 0,6 . From Kapranov's construction, there is a blow-down mapM 0,6 → P 3 .…”

Section: Proof (Of Lemma 8) We Want To Show Thatmentioning

confidence: 99%

“…Fix four distinct points (3, 4, 5, 6) on a smooth genus 0 curve C , and let two points 1, • vary on it. This defines a test surface for (12) We are now in a position to compute the matrix associated to φ * .…”

Section: Proof (Of Lemma 8) We Want To Show Thatmentioning

confidence: 99%

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The Class of the Bielliptic Locus in Genus 3

Faber

Pagani

2014

International Mathematics Research Notices

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Let the bielliptic locus be the closure in the moduli space of stable curves of the locus of smooth curves that are double covers of genus 1 curves. In this paper, we compute the class of the bielliptic locus inM 3 in terms of a standard basis of the rational Chow group of codimension-2 classes in the moduli space. Our method is to test the class on the hyperelliptic locus: this gives the desired result up to two free parameters, which are then determined by intersecting the locus with two surfaces inM 3 . The Main ResultA smooth bielliptic curve is a genus g curve that is a double cover of a smooth genus 1 curve, with 2g − 2 ramification points by the Riemann-Hurwitz formula. It follows that the locus of bielliptic curves has codimension g − 1 inM g , the moduli stack of stable curves of genus g. We are interested in the following general problem: given a family over a base of dimension g − 1, how many bielliptic curves occur in it? We solve this problem by expressing the class of the bielliptic locus in terms of standard classes in the case

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“…[HMu,Harr,EH,Far,Lo,V,CT,CC2]. In contrast, little is known about higher codimension cycles on M g,n , in part because their positivity properties are not as well-behaved.…”

Section: Introductionmentioning

confidence: 99%

Extremal higher codimension cycles on moduli spaces of curves

Chen¹,

Coskun²

2015

Proc. London Math. Soc.

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Abstract. We show that certain geometrically defined higher codimension cycles are extremal in the effective cone of the moduli space Mg,n of stable genus g curves with n ordered marked points. In particular, we prove that codimension two boundary strata are extremal and exhibit extremal boundary strata of higher codimension. We also show that the locus of hyperelliptic curves with a marked Weierstrass point in M3,1 and the locus of hyperelliptic curves in M4 are extremal cycles. In addition, we exhibit infinitely many extremal codimension two cycles in M1,n for n ≥ 5 and in M2,n for n ≥ 2.

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A Counterexample to Fulton's Conjecture on [formula]0,n

Abstract: We describe an elementary counterexample to a conjecture of Fulton on the set of effective divisors on the space M 0 n of marked rational curves.  2002 Elsevier Science (USA)

Cited by 37 publications

References 1 publication

Mori dream spaces

Mori dream spaces

The Class of the Bielliptic Locus in Genus 3

Extremal higher codimension cycles on moduli spaces of curves

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