# Convexity (algebraic geometry)

In algebraic geometry, convexity is a restrictive technical condition for algebraic varieties originally introduced to analyze Kontsevich moduli spaces ${\overline {M}}_{0,n}(X,\beta )$ in quantum cohomology.$^{:§1}$ These moduli spaces are smooth orbifolds whenever the target space is convex. A variety $X$ is called convex if the pullback of the tangent bundle to a stable rational curve $f:C\to X$ has globally generated sections. Geometrically this implies the curve is free to move around $X$ infinitesimally without any obstruction. Convexity is generally phrased as the technical condition
    $H^{1}(C,f^{*}T_{X})=0$
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 $p,q$ in a convex set $C\subset \mathbb {R} ^{n}$, all of the points $tp+(1-t)q$ are contained in that set. There is a vector field ${\mathcal {X}}_{U_{p}}$ in a neighborhood $U_{p}$ of $p$ transporting $p$ to each point $p'\in \{tp+(1-t)q:t\in [0,1]\}\cap U_{p}$. Since the vector bundle of $\mathbb {R} ^{n}$ is trivial, hence globally generated, there is a vector field ${\mathcal {X}}$ on $\mathbb {R} ^{n}$ such that the equality ${\mathcal {X}}|_{U_{p}}={\mathcal {X}}_{U_{p}}$ holds on restriction.

## Examples

There are many examples of convex spaces, including the following.

### Spaces with trivial rational curves

If the only maps from a rational curve to $X$ are constants maps, then the pullback of the tangent sheaf is the free sheaf ${\mathcal {O}}_{C}^{\oplus n}$ where $n=\dim(X)$. These sheaves have trivial non-zero cohomology, and hence they are always convex. In particular, Abelian varieties have this property since the Albanese variety of a rational curve $C$ is trivial, and every map from a variety to an Abelian variety factors through the Albanese.

### Projective spaces

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
    $0\to {\mathcal {O}}\to {\mathcal {O}}(1)^{\oplus (n+1)}\to {\mathcal {T}}_{\mathbb {P} ^{n}}\to 0$
If we only need to consider degree $d$ embeddings, there is a short exact sequence
    $0\to {\mathcal {O}}_{C}\to {\mathcal {O}}_{C}(d)^{\oplus (n+1)}\to f^{*}{\mathcal {T}}_{\mathbb {P} ^{n}}\to 0$
giving the long exact sequence
    ${\begin{aligned}0&\to H^{0}(C,{\mathcal {O}})\to H^{0}(C,{\mathcal {O}}(d)^{\oplus (n+1)})\to H^{0}(C,f^{*}{\mathcal {T}}_{\mathbb {P} ^{n}})\\&\to H^{1}(C,{\mathcal {O}})\to H^{1}(C,{\mathcal {O}}(d)^{\oplus (n+1)})\to H^{1}(C,f^{*}{\mathcal {T}}_{\mathbb {P} ^{n}})\to 0\end{aligned}}$
since the first two $H^{1}$-terms are zero, which follows from $C$ being of genus $0$, and the second calculation follows from the Riemann–Roch theorem, we have convexity of $\mathbb {P} ^{n}$. Then, any nodal map can be reduced to this case by considering one of the components $C_{i}$ of $C$.

### Homogeneous spaces

Another large class of examples are homogenous spaces $G/P$ where $P$ is a parabolic subgroup of $G$. These have globally generated sections since $G$ acts transitively on $X$, meaning it can take a bases in $T_{x}X$ to a basis in any other point $T_{y}X$, hence it has globally generated sections. Then, the pullback is always globally generated. This class of examples includes Grassmannians, projective spaces, and flag varieties.

### Product spaces

Also, products of convex spaces are still convex. This follows from the Künneth theorem in coherent sheaf cohomology.

### Projective bundles over curves

One more non-trivial class of examples of convex varieties are projective bundles $\mathbb {P} ({\mathcal {E}})$ for an algebraic vector bundle ${\mathcal {E}}\to C$ over a smooth algebraic curve$^{pg6}$.

## Applications

There are many useful technical advantages of considering moduli spaces of stable curves mapping to convex spaces. That is, the Kontsevich moduli spaces ${\overline {M}}_{0,n}(X,\beta )$ have nice geometric and deformation-theoretic properties.

### Deformation theory

The deformations of $f:C\to X$ in the Hilbert scheme of graphs $\operatorname {Hom} (C,X)\subset \operatorname {Hilb} _{C\times X/\operatorname {Spec} (\mathbb {C} )}$ has tangent space
    $T_{\operatorname {Hom} (C,X)}([f])\cong H^{0}(C,f^{*}T_{X})$
where $[f]\in \operatorname {Hom} (C,X)$ is the point in the scheme representing the map. Convexity of $X$ gives the dimension formula below. In addition, convexity implies all infinitesimal deformations are unobstructed.

### Structure

These spaces are normal projective varieties of pure dimension
    $\dim({\overline {M}}_{0,n}(X,\beta ))=\dim(X)+\int _{\beta }c_{1}(T_{X})+n-3$
which are locally the quotient of a smooth variety by a finite group. Also, the open subvariety ${\overline {M}}_{0,n}^{*}(X,\beta )$ parameterizing non-singular maps is a smooth fine moduli space. In particular, this implies the stacks ${\overline {\mathcal {M}}}_{0,n}(X,\beta )$ are orbifolds.

### Boundary divisors

The moduli spaces ${\overline {M}}_{0,n}(X,\beta )$ have nice boundary divisors for convex varieties $X$ given by
    $D(A,B;\beta _{1},\beta _{2})={\overline {M}}_{0,A\cup \{\bullet \}}(X,\beta _{1})\times _{X}{\overline {M}}_{0,B\cup \{\bullet \}}(X,\beta _{2})$
for a partition $A\cup B$ of $[n]$ and $\{\bullet \}$ the point lying along the intersection of two rational curves $C=C_{1}\cup C_{2}$.