Let $X$ be a smooth projective variety. By Definition 24.4.2 in the 2003 book Mirror Symmetry, $X$ is called convex if $h^1\left( \Sigma, f^*T_X \right) = 0$ for every genus zero stable map $f:\Sigma \to X$.

**Question:** Why is a convex variety called convex?

On page 505 of the book, it was said that $H^1\left( \Sigma, f^*T_X \right) $ measures obstructions to the deformations of the map $f$ (when the structure of the source curve $\Sigma$ is fixed). But I don't know what the obstructions mean, or what zero obstruction means. So I'm also asking the following

**Related question:** What do zero and nonzero obstructions to the deformations of the map $f$ mean?

**My background and what answer I'm looking for:** I have close-to-zero background in deformation theory. So I don't expect to fully understand the precise mathematical meaning over one night. I would appreciate it very much if someone could explain what zero and nonzero obstructions mean in an intuitive way. If this can be further supported by concrete examples for zero and nonzero obstructions, then it will be perfect.