__NOTOC__ In algebraic geometry, a Barth surface is one of the complex nodal surfaces in 3 dimensions with large numbers of double points found by . Two examples are the Barth sextic of degree 6 with 65 double points, and the Barth decic of degree 10 with 345 double points. For degree 6 surfaces in P³, showed that 65 is the maximum number of double points possible. The Barth sextic is a counterexample to an incorrect claim by

Francesco Severi Francesco Severi (13 April 1879 – 8 December 1961) was an Italian mathematician. He was the chair of the committee on Fields Medal on 1936, at the first delivery. Severi was born in Arezzo, Italy. He is famous for his contributions to algebr ...

in 1946 that 52 is the maximum number of double points possible.

Informal accounting of the 65 ordinary double points of the Barth Sextic

The Barth Sextic may be visualized in three dimensions as featuring 50 finite and 15 infinite ordinary double points (nodes). Referring to the figure, the 50 finite ordinary double points are arrayed as the vertices of 20 roughly

tetrahedral In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...

shapes oriented such that the bases of these four-sided "outward pointing" shapes form the triangular faces of a regular

icosidodecahedron In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 ...

. To these 30 icosidodecahedral vertices are added the summit vertices of the 20 tetrahedral shapes. These 20 points themselves are the vertices of a concentric regular dodecahedron circumscribed about the inner icosidodecahedron. Together, these are the 50 finite ordinary double points of the figure. The 15 remaining ordinary double points at infinity correspond to the 15 lines that pass through the opposite vertices of the inscribed icosidodecahedron, all 15 of which also intersect in the center of the figure. .

References

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External links

* * * *{{cite web, url=http://cage.rug.ac.be/~hs/barth/barth.html , archive-url=https://web.archive.org/web/20080125161923/http://cage.rug.ac.be/~hs/barth/barth.html , url-status=dead , archive-date=2008-01-25 , title=Animations of Barth surfaces Algebraic surfaces Complex surfaces

Informal accounting of the 65 ordinary double points of the Barth Sextic

See also

References

External links