# 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 modeling  and in the complex classification of higher dimensional varieties .

Suppose that X is an embedded real 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 and obtained the following corollary:

• A vertex in G(X) has label ≤3 and an edge in G(X) has label ≤8. If an edge has label 3 or 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 result from  and  give a characterization of the unit-sphere:

• If G(X) has a vertex with label ≥3, then the linear normalization of X is biregular isomorphic to a projective quadric whose real points form the unit-sphere:
{(x:y:z:w)|-w^2+x^2+y^2+z^2=0}.

The above examples of simple family graphs illustrate hexagonal webs of curves in simple families that are characterized in  as follows:

• 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.

## Hexagonal webs

Let W be a set of curves on the surface X and let Wp denote the subset of curves in W that contain the point p.

Let G(W) be the graph with vertex set {p in X: |Wp|=3} and labeled edge set {({v,w},C): C ∈ W and v, w ∈ C are pairwise distinct}.

We call W a hexagonal web if the vertex set of G(W) is not contained in some reducible curve and if a general edge {p,q} of G(W) is contained in the following subgraph where A, B, C, D, E, F, G, H and I are pairwise distinct curves in W.

If W is a hexagonal web, then the topological procedure illustrated below results in a closed hexagon.  Choose a general curve C in W and two general points p and q that lie on C. Draw all the curves in W that contain either p or q and obtain at least two new intersection points r and s.  Draw all the curves in W that contain r and s and obtain at least two new intersection points. We repeat the last step and 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, Journal of the London Mathematical Society, 2021, [journal], [arxiv]