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A septic with 99 real nodes. (English) Zbl 1112.14043

The author improves on the previously known lower bound for the maximal number of nodes of a surface of degree 7 in \({\mathbb P}^3\) by exhibiting one with 99 nodes (the previous best was 93). Moreover, the 99 nodes are all real and the author has written computer code to draw pictures of the surface. In characteristic 5 he can find a surface with a hundredth node.
The surface concerned is completely explicit. Its equation is \[ \begin{split} x^7-21x^5y^2+35x^3y^4-7xy^6+7z(x^2+y^2) (x^4+2x^2y^2+y^4-8x^2z^2-8y^2z^2+16z^4) \\ -(z+a_5w)((x^2+y^2) (z+w)+a_1z^3+a_2z^2w+a_3zw^2+a_4w^3)^2=0 \end{split} \]
where \(49a_1=-84\alpha^2-384\alpha-56\), \(49a_2=-224\alpha^2+24\alpha-28\), \(49a_3=-196\alpha^2+24\alpha-196\), \(49a_4=-56\alpha^2-8\alpha-56\) and \(a_5=49\alpha^2-7\alpha+50\), and \(\alpha\approx -0.14011\) is the unique real root of \(7t^3+7t+1=0\). (Using the non-real roots also gives 99 nodes, but not real ones.) The surface has dihedral symmetry group \(D_7\), and the singularities of a septic surface \(S\) having such symmetries are the orbits under this group of the nodes of the curves \(S_y=S\cap\{y=0\}\). To find a surface with 99 nodes one therefore needs a plane curve of degree 7 with 15 nodes and equation of a suitable form. The author guides himself in his search for such curves by first finding, by exhaustion, all such curves defined over some small finite fields (he gives examples over \({\mathbb F_{19}}\), \({\mathbb F_{31}}\), \({\mathbb F_{43}}\) and \({\mathbb F_{53}}\) among others). He observes some features of the geometry of such curves and chooses suitable lifts to characteristic zero. The proofs also rely heavily on computation.

MSC:

14J17 Singularities of surfaces or higher-dimensional varieties
14Q10 Computational aspects of algebraic surfaces
14G15 Finite ground fields in algebraic geometry
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References:

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