# Moduli spaces of horizontal curves

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.

• 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. Aug 25, 2021 at 15:11

## 1 Answer

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 $$y=\infty$$.