In algebraic geometry, convexity is a restrictive technical condition for algebraic varieties originally introduced to analyze Kontsevich moduli spaces
is called convex if the pullback of the tangent bundle to a stable rational curve
[2] Geometrically this implies the curve is free to move around
Convexity is generally phrased as the technical condition since Serre's vanishing theorem guarantees this sheaf has globally generated sections.
Intuitively this means that on a neighborhood of a point, with a vector field in that neighborhood, the local parallel transport can be extended globally.
This generalizes the idea of convexity in Euclidean geometry, where given two points
is trivial, hence globally generated, there is a vector field
There are many examples of convex spaces, including the following.
If the only maps from a rational curve to
These sheaves have trivial non-zero cohomology, and hence they are always convex.
[4] Projective spaces are examples of homogeneous spaces, but their convexity can also be proved using a sheaf cohomology computation.
Recall the Euler sequence relates the tangent space through a short exact sequence If we only need to consider degree
embeddings, there is a short exact sequence giving the long exact sequence since the first two
Then, any nodal map can be reduced to this case by considering one of the components
Another large class of examples are homogenous spaces
This class of examples includes Grassmannians, projective spaces, and flag varieties.
This follows from the Künneth theorem in coherent sheaf cohomology.
One more non-trivial class of examples of convex varieties are projective bundles
over a smooth algebraic curve[3]pg 6.
There are many useful technical advantages of considering moduli spaces of stable curves mapping to convex spaces.
That is, the Kontsevich moduli spaces
have nice geometric and deformation-theoretic properties.
in the Hilbert scheme of graphs
is the point in the scheme representing the map.
gives the dimension formula below.
In addition, convexity implies all infinitesimal deformations are unobstructed.
[5] These spaces are normal projective varieties of pure dimension which are locally the quotient of a smooth variety by a finite group.
parameterizing non-singular maps is a smooth fine moduli space.
have nice boundary divisors for convex varieties
the point lying along the intersection of two rational curves