Webs of rational curves on surfaces

Simple family graphs

Lines play a central role in classical geometry and have been further developed as geodesics in Riemannian geometry. From an algebro geometric point of view, we can consider lines as rational curves of minimal degree. Such "simple" curves play an important role in geometric modelling  and in the complex classification of higher dimensional varieties .

Suppose that X is an embedded algebraic surface.

A simple family is an algebraic family of minimal degree rational curves that covers X such that a general curve in this family is smooth outside the singular locus of X. Moreover, we assume that the dimension of a simple family is as large as possible.

The intersection product of two simple families that cover X is defined as the number of intersections between a general curve in the first family and a general curve in the second family, outside the singular locus of X.

The simple family graph G(X) is defined as follows:

• Each vertex is a simple family of X. A vertex is labeled with the dimension of the simple family.
• We draw between two simple families an edge if their intersection product is at least two. An edge is labeled with the intersection product.

The vertices in the following examples of simple family graphs are colored according to their corresponding simple families.            The simple family graph of the Clebsch surface has 27 vertices and all edges are labeled 2.

In  we classified simple family graphs of real embedded surfaces. A vertex in G(X) has label ≤3 and an edge in G(X) has label ≤8. If an edge has label ≥5, then G(X) contains ≤2160 vertices and ≤2262600 edges.

What properties of X are encoded in its simple family graph G(X)? The following theorem gives a characterization of the unit-sphere and of hexagonal webs of curves in simple families.

The linear normalization LN(X) is the surface whose coordinate ring is the integral closure of the coordinate ring of X. LN(X) is a surface in projective space and X is (an affine chart of) a linear projection of LN(X). The linear normalization LN(S) of the unit-sphere S is
{(x:y:z:w)|-w^2+x^2+y^2+z^2=0}.

Theorem [ 3 , 4 ].

• If G(X) has a vertex with label at least three, then LN(X) is biregular isomorphic to LN(S), where S is the unit-sphere.
• If three vertices in G(X) do not share an edge, then their corresponding three simple families are 1-dimensional and form a hexagonal web.

The 2nd, 3th, 5th and 6th example of the above simple family graphs illustrate hexagonal webs that are defined by three vertices that do not share an edge.

Hexagonal webs

Let W be a set of curves on surface X such that through almost every point in X pass exactly 3 curves in W. We call W a hexagonal web if the topological procedure illustrated below results in a closed hexagon.  Fix a small neighborhood U around a general point p in X. Choose point q near p on one of the 3 curves in W.  Points r and s in U are intersections of curves in W that pass through q or p. Repeat the procedure for r and s instead of q.  If the resulting hexagon closes, then W is a hexagonal web. If the outcome is not a closed hexagon, then W is not a hexagonal web.

A discrete realization of a hexagonal web of simple curves results in a nice triangularization of the underlying surface and is of interest in geometric modeling.  References

  Darboux cyclides and webs from circles H. Pottmann, L. Shi and M. Skopenkov, Computer Aided Geometric Design, 29(1):77-79, 2012, [arxiv]  Mori geometry meets Cartan geometry: Varieties of minimal rational tangents J.M. Hwang, Proceedings of ICM2014, [arxiv]  Minimal degree rational curves on real surfaces N. Lubbes, Advances in Mathematics, 345:263-288, 2019, [journal], [arxiv]  Webs of rational curves on real surfaces and a classification of real weak del Pezzo surfaces N. Lubbes, 2018, [arxiv]