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Let $f:X\rightarrow Y$ be a morphism of projective varieties. We may assume that $X$ and $Y$ are smooth, and $f$ is flat of relative dimension one. Fix an ample divisor $A$ on $X$.

I would like to ask if there exists a compact moduli space $\overline{M}_{g,d}(X)$ such that all points of $\overline{M}_{g,d}(X)$ represent curves $C\subset X$ of arithmetic genus $g$ and degree $d$ (with respect to $A$) with the following additional property:

$(\star)$ none of the irreducible components of $C$ is contracted by $f$.

Thank you.

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  • $\begingroup$ As noted by@WillSawin, there is no fine module space for such curves. However, when $g$ equals zero, there is the space of quasi maps. This has some of the properties that you list. $\endgroup$ Aug 25, 2021 at 15:11

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It is not possible to have such a moduli space that contains all the smooth curves of genus $g$ and degree $d$ and over which the universal family of curves is proper.

Let $X = \mathbb P^1 \times \mathbb P^1$ with coordinates $x,y$, $Y= \mathbb P^1$, $f$ the projection onto the $x$ coordinate. Let $C_t$ be given by $y=tx$. In the limit as $t \to \infty$, this converges to the curve with the vertical component $x=0$ (in addition to the horizontal component $y=\infty$). So if the moduli space is compact and the universal family is proper, the limit as $t \to \infty$ of $C_t$ will contain that component.

You could try to do something non-proper but if you want the universal family to be flat you will still have problems, as then $C_{\infty}$ will have to be contained in the union of $x=0$ and $y = \infty$, so to have the same degree as $C_t$ according to an ample line bundle will have to contain both $x=0$ and $y=\infty$.

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