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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

MSC:
 14J30 $$3$$-folds 14N10 Enumerative problems (combinatorial problems) in algebraic geometry 14B05 Singularities in algebraic geometry
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References:
 [1] Arnold, V.I., Gusein-Zade, S.M., Varchenco, A.N.: Singularities of differentiable maps, vol. I. Basel Boston Stuttgart: Birkhäuser 1985 [2] Artin, M.: On the solutions of analytic equations. Invent. Math.5, 277-291 (1968) · Zbl 0172.05301 [3] Burkhardt, H.: Grundzüge einer allgemeinen Systematik der hyperelliptischen Funktionen I. Ordnung. Math. Ann.38, 161-224 (1891) · JFM 23.0490.01 [4] Brieskorn, E., Knörrer, H.: Ebene algebraische Kurven. Basel Boston Stuttgart: Birkhäuser 1981 · Zbl 0508.14018 [5] Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces. Erg der Math. (3), 4. Berlin Heidelberg New York: Springer 1984 · Zbl 0718.14023 [6] Dolgacev, I.: Cohomologically insignificant degenerations of algebraic varieties. Compos. Math.42, 279-313 (1981) · Zbl 0466.14003 [7] Friedman, R.D.: Simultaneous resolution of threefold double points. Math. Ann.274, 671-689 (1986) · Zbl 0576.14013 [8] Fulton, W.: Intersection theory. Erg. der Math. (3),2. Berlin Heidelberg New York: Springer 1984 · Zbl 0541.14005 [9] Kalker, A.A.C.M.: Cubic fourfolds with fifteen ordinary double points. Thesis Leiden (1986) [10] Miyaoka, Y.: The maximal number of quotient singularities on surfaces with given numerical invariants. Math. Ann.268, 159-171 (1984) · Zbl 0534.14019 [11] Mitchell, H.H.: Determination of all primitive collineation groups in four variables which contain homologies. Amer. J. Math.36, 1-12 (1914) · JFM 45.0253.01 [12] Shephard, G.C., Todd, J.A.: Finite unitary reflection groups. Canad. J. Math.VI, 274-304 (1954) · Zbl 0055.14305 [13] Varchenco, A.N.: On the semicontinuity of the spectrum and an upper bound for the number of singular points of projective hypersurfaces. Dokl. Akad. Nauk. U.S.S.R.270(6) (1983) [14] Zariski, O.: Algebraic surfaces. Second edition. Erg. Math.61. Berlin Heidelberg New York: Springer 1971 · Zbl 0219.14020
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