A singular cubic outside the lattice family

(The discriminant vanishes.) For the coefficient pair $(g_2,g_3)=(3,1)$ the number displayed in the statement is $\mathrm{\Delta }=g_2^3-27g_3^2=27-27=0$, since $3^3=27$ and $27\cdot 1^2=27$.

(The point lies on the affine curve.) With the chart coordinates of [F5], the cubic of the statement has affine equation $y^2=4x^3-3x-1$ at $Z=1$; writing $F(x,y):=y^2-(4x^3-3x-1)$, at $p=(-1\mathrm{/}2,0)$ one has $x^2=1\mathrm{/}4$ and $4x^3-3x-1=4(-1\mathrm{/}8)-3(-1\mathrm{/}2)-1=-1\mathrm{/}2+3\mathrm{/}2-1=0$, so $F(p)=0-0=0$ and $p$ lies on the affine curve.

(The differential vanishes at the point.) The partial derivatives of $F$ are $\mathrm{\partial }F\mathrm{/}\mathrm{\partial }y=2y$ and $\mathrm{\partial }F\mathrm{/}\mathrm{\partial }x=-(12x^2-3)$; at $p$ these are $2\cdot 0=0$ and $-(12\cdot \frac{1}{4}-3)=-(3-3)=0$. Thus $dF(p)=0$, so the plane curve $F=0$ has vanishing differential at $p$; this is the elementary singularity criterion of the affine chart.

(Factorisation and the unit square root.) Expanding $(x-1)(2x+1{)}^2=(x-1)(4x^2+4x+1)=4x^3-3x-1$ gives $4x^3-3x-1=(x-1)(2x+1{)}^2$. Put $s:=x+\frac{1}{2}$, so that $2x+1=2s$ and $x-1=s-\frac{3}{2}$, and hence $4x^3-3x-1=(s-\frac{3}{2})(2s{)}^2=4s^3-6s^2=s^{2}h(s)$ with $h(s):=4s-6$. Apply [F3] to $G(s,w):=w^2-h(s)=w^2-4s+6$ at $(0,w_0)$ with $w_0:=i\sqrt{6}$: $G(0,w_0)=-6+6=0$ and $\mathrm{\partial }G\mathrm{/}\mathrm{\partial }w(0,w_0)=2w_0=2i\sqrt{6}\mathrm{\ne }0$; hence there is a holomorphic $u$ on a disc around $0$ with $u(0)=w_0$ and $u(s{)}^2=h(s)=4s-6$. By [F4] there is $\epsilon >0$ with $u(s)\mathrm{\ne }0$ for $\mathrm{\mid }s\mathrm{\mid }<\epsilon$.

(Two smooth branches crossing at $p$.) With $u$ as in step 1.4, the identity $y^2-s^{2}h(s)=(y-su(s))(y+su(s))$ exhibits the affine curve near $p$ (which is $s=0$, $y=0$) as the union of the two graphs ${\mathrm{\Sigma }}_{\pm }=\{(s,y):y=\pm su(s)\}$ over the $s$-coordinate, $\mathrm{\mid }s\mathrm{\mid }<\epsilon$. Each ${\mathrm{\Sigma }}_{\pm }$ is smooth with parametrisation $s\mapsto (s,\pm su(s))$, and the two branches meet exactly at $s=0$: for $0<\mathrm{\mid }s\mathrm{\mid }<\epsilon$ one has $su(s)\mathrm{\ne }0$ by step 1.4, so the two points $(s,su(s))$ and $(s,-su(s))$ are distinct. The tangent directions at the meeting point are $(1,w_0)$ and $(1,-w_0)$ with $w_0\mathrm{\ne }0$, hence distinct, so the branches cross transversally. Moreover $\phi (s):=su(s)$ satisfies $\phi (0)=0$ and ${\phi }^{\mathrm{\prime }}(0)=u(0)=w_0\mathrm{\ne }0$; applying [F3] to $(y,s)\mapsto \phi (s)-y$ at $(0,0)$ gives a holomorphic inverse branch $\psi$ with $\phi (\psi (y))=y$ for small $y$. The inverse branches of the two curve graphs are $s=\psi (y)$ and $s=\psi (-y)$.

(The point is not a holomorphic graph in either direction.) Let $P=D_s\times D_y$ be any small polydisc around $(0,0)$ contained in the domain of $u$ and $\psi$, with $u$ nowhere zero on $D_s$ and $\phi (D_s)\subset D_y$. (i) For small $0\mathrm{\ne }s\in D_s$, both $(s,su(s))$ and $(s,-su(s))$ are points of the curve in $P$ with the same $s$-coordinate and distinct $y$-coordinates; a graph over the $s$-coordinate would contain exactly one point over $s$, so the curve is not a holomorphic graph over $s$. (ii) For small $0\mathrm{\ne }y\in D_y$ with $\psi (\pm y)\in D_s$, the points $(\psi (y),y)\in {\mathrm{\Sigma }}_{+}$ and $(\psi (-y),y)\in {\mathrm{\Sigma }}_{-}$ are distinct points of the curve in $P$ with the same $y$-coordinate, because $\psi$ is injective on $D_y$ and $y\mathrm{\ne }-y$; so the curve is not a holomorphic graph over $y$ either. By [F2] a Jacobian-rank nonsingular point of a plane curve germ is a holomorphic graph over one of the two coordinates, so $p$ is not nonsingular in the Jacobian-rank sense.

(No lattice has these invariants.) Suppose a full complex lattice $\mathrm{\Lambda }$ had invariants $g_2=3$, $g_3=1$. Then its associated cubic $C_{\mathrm{\Lambda }}$ of [F1] is exactly the projective cubic of the statement, and [F1] asserts that $C_{\mathrm{\Lambda }}$ is nonsingular in the Jacobian-rank sense at every point. But the affine point $p$ is a point of $C_{\mathrm{\Lambda }}$ by step 1.2 and is not Jacobian-rank nonsingular by step 3.1, a contradiction. The same conclusion is visible in the numbers alone: [F1] gives $\mathrm{\Delta }(\mathrm{\Lambda })\mathrm{\ne }0$, while step 1.1 computes $\mathrm{\Delta }=0$ for the pair $(3,1)$. Hence no full complex lattice realizes the invariants $(3,1)$, so the cubic of the statement is a degeneration outside the lattice family.

(Assembly.) Steps 1.1, 1.2 and 2.1 show that the projective cubic $Y^{2}Z=4X^3-3X{Z}^2-Z^3$ has $\mathrm{\Delta }=3^3-27\cdot 1^2=0$ and has at $(x,y)=(-1\mathrm{/}2,0)$ an affine singular point at which the two smooth branches $y=\pm (x+\frac{1}{2})u(x+\frac{1}{2})$ cross transversally, with vanishing differential recorded in step 1.3; step 4.1 shows that this coefficient pair is excluded for every full lattice, by both the nonsingularity clause and the nonvanishing-discriminant clause of [F1]. This is the asserted degeneration. ∎