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In algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, a Togliatti surface is a nodal surface
In algebraic geometry, a nodal surface is a surface in (usually complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex ...
of degree five with 31 nodes
In general, a node is a localized swelling (a "knot") or a point of intersection (a Vertex (graph theory), vertex).
Node may refer to:
In mathematics
*Vertex (graph theory), a vertex in a mathematical graph
*Vertex (geometry), a point where two ...
. 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 be optimal.
See also
*Barth surface
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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 degre ...
*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 ...
*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 Qu ...
References
*. *.External links
* * Algebraic surfaces Complex surfaces {{algebraic-geometry-stub