Singular Fibers in Elliptic Fibrations on the Rational Elliptic Surface

Stipsicz, András I.; Szabó, Zoltán; Szilárd, Ágnes

doi:10.1007/s-10998-007-2137-2

Cited by 5 publications

(18 citation statements)

References 9 publications

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“…The parameters appearing in the above lists have geometric meaning: basically they encode the base locus of an elliptic pencil on the Hirzebruch surface F 2 , see (20), (21), (22), (23). For example, a + , a − (and similarly b + , b − ) determine the locations of the base points on the fibers of the Hirzebruch surface, see also Figures 4 and 5. In order to state our results, it will be useful to consider some polynomial expressions M, L, A and B of these parameters:…”

Section: The Precise Versions Of the Main Theoremmentioning

confidence: 99%

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Hitchin fibrations on moduli of irregular Higgs bundles and motivic wall-crossing

Ivanics

Stipsicz

Szabó

2019

Journal of Pure and Applied Algebra

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In this paper we give a complete description of the Hitchin fibration on all 2-dimensional moduli spaces of rank-2 irregular Higgs bundles with two poles on the projective line. We describe the dependence of the singular fibers of the fibration on the eigenvalues of the Higgs fields, and describe the corresponding motivic wall-crossing phenomenon in the parameter space of parabolic weights.Assumption 1.1. At least one (equivalently, the generic) fiber of h is a smooth elliptic curve.The irregular Hitchin map h extends to a map h∶ M ss ∪ E ∞ → CP 1 , where E ∞ is a complex curve, which is the fiber at infinity of the fibration h. It turns out that h is

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Section: The Precise Versions Of the Main Theoremmentioning

confidence: 99%

“…For the local forms of the Higgs fields that we will consider, see (20), (22), (51) and (53) (regular semisimple case) and (21), (23), (52) and (54) (non-semisimple case).…”

Section: Preparatory Materialsmentioning

confidence: 99%

Hitchin fibrations on moduli of irregular Higgs bundles and motivic wall-crossing

Ivanics

Stipsicz

Szabó

2019

Journal of Pure and Applied Algebra

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“…The spectral curve (χ θ (ζ) = 0) has an expansion near t i in which these parameters have a geometric meaning. The lowest index matrices A n , B n and A n,i in Equations (7), (8) and (9) encode the base locus of a pencil.…”

Section: Preliminariesmentioning

confidence: 99%

“…Various possibilities of singular fibers in an elliptic fibration on a rational elliptic surface was analyzed in detail in [5,7,8]. In particular, we have Section 6]) Suppose that π∶ X → CP 1 is an elliptic fibration on a rational elliptic surface X.…”

Section: Singular Fibers In Elliptic Fibrationsmentioning

confidence: 99%

“…A simple argument shows that the fiber at infinity is either I * 1 (when the polar part is untwisted at the pole of multiplicity 2) or I * 2 (when the polar part is twisted at the pole of multiplicity 2). Since by [8,Section 6] elliptic fibrations on the rational elliptic surface with a singular fiber I * 1 or I * 2 have at least three singular fibers (cf. Corollary 3.2), we conclude Theorem 1.5.…”

Section: Introductionmentioning

confidence: 99%

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