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In algebraic geometry, a Barth surface is one of the complex
nodal surface
In algebraic geometry, a nodal surface is a surface in (usually complex) projective space whose only singularities are nodes. A major problem about them is to find the maximum number of nodes of a nodal surface of given degree.
The following t ...
s 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 roughlytetrahedral
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
A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, ...
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. .
See also
*Endrass surface
In algebraic geometry, an Endrass surface is a nodal surface of degree 8 with 168 real nodes, found by . , it remained the record-holder for the most number of real nodes for its degree; however, the best proven upper bound, 174, does not match t ...
*Sarti surface
In algebraic geometry, a Sarti surface is a degree-12 nodal surface with 600 nodes, found by . The maximal possible number of nodes of a degree-12 surface is not known (as of 2015), though Yoichi Miyaoka showed that it is at most 645.
Sarti has ...
*Togliatti surface
In algebraic geometry, a Togliatti surface is a nodal surface of degree five with 31 nodes. The first examples were constructed by . proved that 31 is the maximum possible number of nodes for a surface of this degree, showing this example to ...
*List of algebraic surfaces
This is a list of named algebraic surfaces, compact complex surfaces, and families thereof, sorted according to their Kodaira dimension following Enriques–Kodaira classification.
Kodaira dimension −∞
Rational surfaces
* Projective plane Qua ...
References
*. *. *.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