For positive integers $n$, $g$ and $d$, the moduli space $M(n,g,d)$ of degree d holomorphic maps to $\mathbb{CP}^n$ from non-singular projective curves of genus g is smooth and irreducible for $d > 2g-2.$ It is contained as an open subset within the compact moduli space $K(n,g,d)$ of “stable maps”, i.e., degree d holomorphic maps to $\mathbb{CP}^n$ from at-worst-nodal projective curves of arithmetic genus $g.$ An unfortunate feature of this very natural compactification is that $M(n,g,d)$ is far from being dense in $K(n,g,d)$. Concretely, this means that many stable maps are not “smoothable”, i.e., they don’t arise as limits of non-singular ones. In my talk, I will explain this phenomenon and a new sufficient condition for smoothability of stable maps, obtained in joint work with Fatemeh Rezaee.

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Last updated: 20 May 2024