Fibrations by non-smooth projective curves of arithmetic genus two in characteristic two

Cañate, Alejandro Simarra; Stöhr, Karl-Otto

doi:10.1016/j.jpaa.2016.02.014

Cited by 2 publications

(3 citation statements)

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“…Therefore such phenomenon can occur only when we are working over fields of characteristic two, three or five. More precisely, it is shown in [2] and [4] that absolutely elliptic curves can occur only in characteristic two or three. In this work we will focus only in the case of characteristic three.…”

Section: Absolutely Elliptic Regular Curves Of Genus Twomentioning

confidence: 99%

“…Several papers studied the classification of fibrations by singular curves from the birational perspective as [29], [30], [23], [24] and [4]. All of them used the well known strategy -that in this context appeared at first in the work of Stöhr (see [29]) andwhich we outline below.…”

Section: Introductionmentioning

confidence: 99%

“…• Λ 3 = Q + S , R 1 + 2R 2 , with Q + S and R 1 + 2R 2 as in (I 2 , (a)) and (I *2 , (a)), respectively, where the irreducible conic curve Q is tangent to the line R 1 at the intersection pointP (3) 1 of R 1 and R 2 , Q intersects R 2 at another point P (3)2 , while the line S intersects R 2 at a third point P need the following restriction: S must be chosen in a way that the tangent line to Q at P In this way have another member Q + S of Λ 3 as in (I 2 , (a)) where the line S is the tangent line to Q at P curve Q is the one that intersects Q at P(3) 1 with index 4 and is tangent to S at P (The configuration of generators and base points can be seen in Figure 23. • Λ 4 = D , Q 1 + S 1 , with D and Q 1 + S 1 as in (I 2 , (b)) and (I * 2 , (c)), respectively, where the nodal cubic curve D intersects the irreducible conic Q 1 with index 4 at P (4) 1 , the tangency point of Q 1 and S 1 ; the node P (4) 2 of D is in Q 1 ; S 1 intersects D at another point P (We need the following restrictions: the point P (4) 1in D must be chosen in a way that both tangent lines to Q 1 at P Thus we must have a member of Λ 4 as in (I 2 , (a)) where the conic curve Q is the one intersecting D and Q 1 at P (4) 1 with indices 6 and 4, respectively, and the line S is the tangent line to Q 1 at P(4)…”

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confidence: 99%

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On the classification of fibrations by genus two singular curves via fibrations by elliptic curves on surfaces

Rodrigues¹,

Salomão²,

Santos³

2023

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In 1944 Zariski discovered that Bertini's theorem on variable singular points is no longer true when we pass from a field of characteristic zero to a field of positive characteristic. In other words, he found fibrations by singular curves, which only exist in positive characteristic. Such fibrations are connected with many interesting phenomena. For instance, the extension of Enrique's classification of surfaces to positive characteristic (Bombieri and Mumford in 1976), the counterexamples of Kodaira vanishing theorem (Mukai in 2013 and Zheng in 2016) and the isolated singularities with infinity Milnor number (Hefez, Rodrigues and Salomão in 2019). In this work we are going to show that the smoothing process introduced by Shimada in 1991 can be used to classify the set of fibrations by genus two singular curves, up to isomorphism among their generic fibers, such that their smoothing are elliptic fibrations on rational surfaces. Moreover we will also describe the vector fields that can be used to recover such fibrations by singular curves via quotient of rational elliptic surfaces.

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Section: Absolutely Elliptic Regular Curves Of Genus Twomentioning

confidence: 99%