論文の公開元へSEMISTABLE FIBRATIONS OVER AN ELLIPTIC CURVE WITH ONLY ONE SINGULAR FIBRE
概要
with connected fibres φ : X → B, where B is a smooth curve. It is well known that if
g(B) ≤ 1 and the fibration is not isotrivial (i.e. such that not all smooth fibres are isomorphic)
then it has at least one singular fibre (cf. [10, Th´eor`eme 4], [9]).
In this short note we describe a construction of semistable fibrations over an elliptic curve
with one unique singular fibre and we give effective examples (section 2). We also establish
some general properties of such fibrations (section 1). Note that the existence of these
semistable fibrations contrasts with Remark 3 in ([9]) which states that any fibration over
an elliptic curve has at least two singular fibres. This claim is said to be an immediate
consequence of arguments used in the proof of Theorem 4 of ([9]). However it is not clear
how those arguments are used. On the other hand there exist fibrations of genus 3 over an
elliptic curve with a unique singular reduced fibre (cf. [7]).
Our construction starts from constructing certain ramified (non Galois) covers C → E
of elliptic curves using monodromy and showing that under suitable hypothesis the surface
C × C has a fibration φ : C × C → E as required.
Notations and conventions. We work over the complex numbers. ∼ denotes numerical
equivalence of divisors, whilst ≡ denotes linear equivalence. The composition of permutations is done right to left.
2. Generalities for a fibration over an elliptic curve with one unique singular fibre
Let E be an elliptic curve with origin OE ∈ E, and let KE be the canonical bundle of
2. Generalities for a fibration over an elliptic curve with one unique singular fibre
E. Let X be a compact smooth complex surface and φ : X → E be a relatively minimal
semistable fibration with general fibre F a curve of genus g ≥ 2.
Note that X is a minimal surface, since any rational curve must be vertical. By the same
reason X is not birational to a ruled surface.
Let ωX|E be the relative dualizing sheaf, and let KX be the canonical divisor of X. De2010 Mathematics Subject Classification. Primary 14H10, 14D06; Secondary 114C20, 14J29. ...
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