Another example also in the flavor of enumerative geometry: by considering the (Deligne—Mumford compactification of) the moduli space of elliptic curves, we see that the point at infinity represents the genus one curve with one node, while each other point represents curves with fixed $j$-invariant. The moduli space now looks somewhat like $\mathbb{P}^1$ (except that it is not, for some stacky reasons, but for current purpose, we can pretend that it is) and we can view the nodal curve as having $j = \infty$. Since any two points on $\mathbb{P}^1$ are equivalent as divisors, the nodal curve is “equivalent” to any other curve with fixed $j$-invariant. (Except when $j$ is $0$, $1$, $1728$.)

This property translates well into the Kontsevich moduli space of stable maps of elliptic curves to say, $\mathbb{P}^2$. The locus of maps of nodal curves are equivalent to the locus of maps of any curves with fixed $j$-invariant as divisors, hence they should have the same enumerative invariants, because after all, enumerative invariants are just intersection numbers on the moduli space of maps. Thus, the problem of counting elliptic curves with fixed $j$-invariant is the same as counting nodal curves, and in $\mathbb{P}^2$, nodal curves are the same as rational curves (as rational plane curves necessarily have nodes if the degree is more than 2). This argument was used by Pandharipande to compute the characteristic numbers of plane elliptic curves with fixed $j$-invariant.

Another example also in the flavor of enumerative geometry: by considering the (Deligne–Mumford compactification of) the moduli space of elliptic curves, we see that the point at infinity represents the genus one curve with one node, while each other point represents curves with fixed $j$-invariant. The moduli space now looks somewhat like $\mathbb{P}^1$ (except that it is not, for some stacky reasons, but for current purpose, we can pretend that it is) and we can view the nodal curve as having $j = \infty$. Since any two points on $\mathbb{P}^1$ are equivalent as divisors, the nodal curve is “equivalent” to any other curve with fixed $j$-invariant. (Except when $j$ is $0$, $1$, $1728$.)

This property translates well into the Kontsevich moduli space of stable maps of elliptic curves to say, $\mathbb{P}^2$. The locus of maps of nodal curves are equivalent to the locus of maps of any curves with fixed $j$-invariant as divisors, hence they should have the same enumerative invariants, because after all, enumerative invariants are just intersection numbers on the moduli space of maps. Thus, the problem of counting elliptic curves with fixed $j$-invariant is the same as counting nodal curves, and in $\mathbb{P}^2$, nodal curves are the same as rational curves (as rational plane curves necessarily have nodes if the degree is more than 2). This argument was used by Pandharipande to compute the characteristic numbers of plane elliptic curves with fixed $j$-invariant.

Another example also in the flavor enumerative geometry: by considering the (Deligne- Mumford compactification of) moduli space of elliptic curves, we see that the point at infinity represents the genus one curve with one node, while each other point represents curve with fix j-invariant. The moduli space now look somewhat like $\mathbb{P}^1$ (except that it is not, for some stacky reasons, but for current purpose, we can pretend that it is  ) and we can view the nodal curve as having $j = \infty$. Since any two points on $\mathbb{P}^1$ are equivalent as divisors  , the nodal curve is "equivalent" to any other curve with fix j-invariant. ( except when j is 0,1,1728)

This property translates well into the Kontsevich moduli space of stable maps of elliptic curves to say, $\mathbb{P}^2$. The locus of maps of nodal curves are equivalent to the locus of maps of any curves with fix j-invariant as divisors, hence they should have the same enumerative invariants, because after all, enumerative invariants are just intersection numbers on the moduli space of maps. Thus, the problem of counting elliptic curves with fix j-invariant is the same as counting nodal curves, and in $\mathbb{P}^2$, nodal curves are the same as rational curves ( as rational plane curves necessarily have nodes if the degree is more than 2  ). This argument was used by Pandharipande to compute the characteristic numbers of plane elliptic curves with fix j-invariant.

Another example also in the flavor of enumerative geometry: by considering the (Deligne—Mumford compactification of) the moduli space of elliptic curves, we see that the point at infinity represents the genus one curve with one node, while each other point represents curves with fixed $j$-invariant. The moduli space now looks somewhat like $\mathbb{P}^1$ (except that it is not, for some stacky reasons, but for current purpose, we can pretend that it is) and we can view the nodal curve as having $j = \infty$. Since any two points on $\mathbb{P}^1$ are equivalent as divisors, the nodal curve is “equivalent” to any other curve with fixed $j$-invariant. (Except when $j$ is $0$, $1$, $1728$.)

This property translates well into the Kontsevich moduli space of stable maps of elliptic curves to say, $\mathbb{P}^2$. The locus of maps of nodal curves are equivalent to the locus of maps of any curves with fixed $j$-invariant as divisors, hence they should have the same enumerative invariants, because after all, enumerative invariants are just intersection numbers on the moduli space of maps. Thus, the problem of counting elliptic curves with fixed $j$-invariant is the same as counting nodal curves, and in $\mathbb{P}^2$, nodal curves are the same as rational curves (as rational plane curves necessarily have nodes if the degree is more than 2). This argument was used by Pandharipande to compute the characteristic numbers of plane elliptic curves with fixed $j$-invariant.