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Let $C$ be an irreducible smooth project curve over $\mathbb C$ and $Y$ a variety over $\mathbb C$ locally of complete intersection. Write $Y^{\text sm}$ for the smooth locus of $Y$. Consider the 'space' of morphisms $f: C \rightarrow Y$ that 'generically take values in $Y^{\text sm}$', i.e. $f(c) \in Y^{\text sm}$ for all but finitely many points $c$ in $C$.

Does anyone know if this 'space' of morphisms is smooth or not? If the answer is yes, could you please suggest some references? If the answer is no, then is it possible to, say, impose extra conditions to make the 'space' smooth? Thank you very much for your help!

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If $C$ has genus $> 2$, even if $C$ has general moduli, even if $Y$ is $\mathbb{P}^n$, typically the Hom scheme, $\text{Hom}(C,\mathbb{P}^n),$ is singular. This sometimes surprises people when they first study Hom schemes, since the Gieseker-Petri theorem is a smoothness theorem that (until you realize what is happening) seems to contradict this phenomenon.

For instance, consider a general curve $C$ of genus $3$. Consider morphisms from $C$ to $\mathbb{P}^2$ whose degree with respect to $\mathcal{O}(1)$ equals $4$. Consider such a morphism $f$ where the image is a line, and where the pullback of $\mathcal{O}(1)$ is the dualizing sheaf.

There is one irreducible component of $\text{Hom}(C,\mathbb{P}^2)$ of dimension $8$ containing $[f]$ that parameterizes morphisms that pullback $\mathcal{O}(1)$ to an invertible sheaf that is isomorphic to the dualizing sheaf. There is a second irreducible component, also of dimension $8$, whose general point parameterizes a morphism with image equal to a line and that pulls back $\mathcal{O}(1)$ be a general invertible sheaf of degree $4$ on $C$. Thus, $\text{Hom}(C,\mathbb{P}^2)$ is reducible at $[f]$, hence it singular at $[f]$.

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  • $\begingroup$ Thank you very much for your answer! Just out of curiosity: can one say anything about the components of $\text{Hom}(C,\mathbb P^2)$? For example, are they smooth? And in general, are the components of $\text{Hom}(C,Y)$ smooth? $\endgroup$
    – user312073
    Jun 9 '17 at 20:21
  • $\begingroup$ "For example, are they smooth?" For fixed $g>1$, $r$ and $d$, for every integer $1\leq s \leq r$, by Gieseker-Petri, for a sufficiently general curve $C$ of genus $g$, the scheme $\mathcal{G}^s_d(C)$ parameterizing linear systems on $C$ of degree $d$ and (projective) dimension $s$ is smooth. There is a Grassmannian bundle over an open subscheme of $\mathcal{G}^s_d(C)$ that embeds into $\text{Hom}^d(C,\mathbb{P}^r)$. Thus, there is a smooth stratification. $\endgroup$ Jun 13 '17 at 8:22

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