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On the Burkhardt quartic. (English) Zbl 0712.14023
The Burkhardt quartic threefold is the subvariety B of \({\mathbb{P}}^ 5(x_ 0:...:x_ 5)\) of equations \(\sigma_ 1 = \sigma_ 4=0\), where \(\sigma_ i\) denotes the i-th symmetric function in \(x_ 0:...:x_ 5\). It is known that B has exactly 45 nodes, which is the maximum number of nodes allowed to a quartic hypersurface in \({\mathbb{P}}^ 4\), according to an inequality due to Varchenko.
In this paper the authors prove that any quartic hypersurface in \({\mathbb{P}}^ 4\) with 45 nodes is projectively equivalent to B. To this aim they use the following special property of any node s in B:
(*) any line intersecting B in s with multiplicity \(\geq 3\) and not contained in B actually meets B in s with multiplicity 4.
Precisely, the authors show, by analyzing the singularities of a certain ramification surface, that if a quartic threefold has 45 nodes, all of them have the property (*). Then the authors prove in two different ways that every quartic threefold with 45 nodes, all of them enjoying the property (*), is projectively equivalent to the Burckhardt quartic: one proof is purely algebraic, the other one is based on the properties of the K3 surfaces.
Reviewer: L.Picco Botta

14J30 \(3\)-folds
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14B05 Singularities in algebraic geometry
Full Text: DOI EuDML
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