Seminar Algebra and Geometry: Andrea Marinatto (Università di Udine)

The Field of Definition of Point-Sets in P^1

Let K be a perfect field of characteristic not equal to two, $\bar{K}$ an algebraic closure of K and let G_K be the Galois group of the extension $\bar{K}/K$.  Let T be a n-point set in $P¹(\bar{K})$. The field of moduli of T is contained in each field of definition but it is not necessarily a field of definition. In this seminar we show that point sets of odd cardinality n≥5 in  $P¹(\bar{K})$ with field of moduli K are defined over their field of moduli. We, also, show that, except for the special case of the 4-point sets, this does not hold in general for point sets of even cardinality n≥6.

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