Recent developments and open problems in  linear series

Bauer, Thomas; Bocci, Cristiano; Cooper, Susan; Rocco, Sandra Di; Dumnicki, Marcin; Harbourne, Brian; Jabbusch, Kelly; Knutsen, Andreas Leopold; Küronya, Alex; Miranda, Rick; Roé, Joaquím; Schenck, Hal; Szemberg, Tomasz; Teitler, Zach

doi:10.4171/114-1/4

Cited by 21 publications

(31 citation statements)

References 51 publications

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“…For type (2) we again assume A = H − E 1 − · · · − E 5 . In this case K ω may be one of the following: it either equals −3H…”

Section: 21mentioning

confidence: 99%

“…Lemma 5.10. Assume A equals any one of the classes of type (1), (2) or (3) specified in Theorem 1.8. If A is represented by a symplectic sphere for some symplectic form ω with K ω = K st , then Theorem 1.8 holds.…”

Section: 21mentioning

confidence: 99%

“…On the one hand, recall the bounded negativity conjecture asserts that any algebraic surface in characteristic zero has C 2 ≥ n X for any prime divisor C ⊂ X and some fixed n X ∈ Z. (for accounts on this conjecture in complex geometry, see for example [20], [3], see also [2] for variations on this conjecture). Note that even for rational manifolds, only certain small ones are known to satisfy this conjecture.…”

mentioning

confidence: 99%

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Stability and existence of surfaces in symplectic 4-manifolds with b ⁺=1

Dorfmeister

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. We establish various stability results for symplectic surfaces in symplectic 4−manifolds with b + = 1. These results are then applied to prove the existence of representatives of Lagrangian ADE-configurations as well as to classify negative symplectic spheres in symplectic 4−manifolds with κ = −∞. This involves the explicit construction of spheres in rational manifolds via a new construction technique called the tilted transport.

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“…For type (2) we again assume A = H − E 1 − · · · − E 5 . In this case K ω may be one of the following: it either equals −3H…”

Section: 21mentioning

confidence: 99%

Section: 21mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Stability and existence of surfaces in symplectic 4-manifolds with b ⁺=1

Dorfmeister

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…Surfaces for which a bound exists are said to have bounded negativity. The Bounded Negativity Conjecture asserts that over the complex numbers every smooth projective surface should have bounded negativity (see [1,5,17] for the history of the conjecture and for results towards the conjecture, and [8] for related boundedness questions).…”

Section: Introductionmentioning

confidence: 99%

Bounded Volume Denominators and Bounded Negativity

Bauer

Harbourne

Küronya

et al. 2019

International Mathematics Research Notices

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In this paper we study the question of whether on smooth projective surfaces the denominators in the volumes of big line bundles are bounded. In particular we investigate how this condition is related to bounded negativity (i.e., the boundedness of self-intersections of irreducible curves). Our first result shows that boundedness of volume denominators is equivalent to primitive bounded negativity, which in turn is implied by bounded negativity. We connect this result to the study of semi-effective orders of divisors: Our second result shows that negative classes exist that become effective only after taking an arbitrarily large multiple.

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“…for generic points in P 2 under assuming the Segre-Harbourne-Gimigliano-Hirschowitz Conjecture, (see [2] for recent account on this conjecture). In this note we study limiting shapes for star configurations in projective spaces of arbitrary dimension.…”

Section: Introductionmentioning

confidence: 99%