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The moduli of rational Weierstrass fibrations over \(\mathbb{P}^ 1\): Isotropy stratification. (English) Zbl 0774.14008

The author considers the moduli space \(W\) of Weierstrass fibrations over \(\mathbb{P}^ 1\) whose section has self-intersection \(-1\) in the associated elliptic surface with the field of complex numbers. A Weierstrass fibration \(\eta^ 2=\xi^ 3+a\xi+b\) corresponds to a point \((a,b)\in V_ 4\times V_ 6\) where \(V_ n\) is the set of binary homogeneous forms of degree \(n\) and, if \(G=\text{GL}_ 2/(\pm I)\), then, for a suitable Zariski open set \(X\) in \(V_ 4\times V_ 6\) satisfying a stability condition, one has \(W=X/G\). As \(X\) can be partitioned using isotropy groups of elements, there is an induced stratification of \(W\), called the isotropy stratification. The isotropy stratification, in particular with respect to “specialization” order, is explicitly determined. The closure of maximal non-principal strata are seen to be the irreducible components of the singular locus of \(W\).

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
14J10 Families, moduli, classification: algebraic theory
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
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