Show that the image in jacobian has an ordinary double point

Question

I'm solving this problem from 11.12 in Birkenhake C., Lange H. - Complex abelian varieties.

Here $W_2$ is an image of $\mu:C^{(2)}\longrightarrow \operatorname{Pic}^2(C)$. Using Rhiemann singularity theorem I've shown that $\operatorname{dim} \operatorname{Sing}W_2=0$ and that it is actually one point. Also by Riemann's Singularity theorem $\operatorname{mult}(l)=h^0(l)=2$. Hence I know that the singularity is a double point, but I have no clue why this is an ordinary double point. So my question is "How to show that this is an ordinary double point?".

I really appreciate any help you can provide.

Answer 1

First solution: You can calculate the self-intersection of the rational curve $E=\mu^{-1}(p)$. If it is $-2$, then the contraction gives an ordinary node. Here is a way to prove it: Let $$f:C\times C\to C^{(2)}$$ be the double cover branched at the diagonal. By definition, the pullback $F=f^{-1}(E)$ is the correspondence curve $\{(x,\tau(x))\in C\times C|x\in C\}$ which is isomorphic to $C$, where $\tau$ is the involution given by canonical mapping.

We first compute the self-intersection of $F$. By adjunction formula

$$K_{F}=(K_{C\times C}\otimes [F])|_{F}.$$

Intersection with $F$ on both sides, using the fact that $K_F\cdot F=\deg(K_F)=4$ and $K_{C\times C}=\pi_1^*K_C\otimes \pi_2^*K_C$ where $\pi_i$ is projection to $i$-th coordinate and the intersection of $F$ with a vertical or horizontal curve is $1$, we get $4=8+F\cdot F$, so $F\cdot F=-4$.

How to compute $E\cdot E$? By projection formula $$f_*F\cdot E=f_*(F\cdot f^*E)$$

together with $f_*F=2E$ and $f^*E=F$ as divisors, we conclude that $E\cdot E=-2$.

Second Solution: Alternatively, Riemann singularities theorem also tells you the projective tangent cone of the singularity at the theta divisor, more precisely from Griffiths and Harris, page 343

The projective tangent cone to $W_d$ at point $\mu(D)$ is the union $$T_{\mu(D)}=\bigcup_{\lambda\in \mathbb P^r} \bar{D}_{\lambda},$$ where $D$ is a degree $d$ effective divisor on $C$ varies in a $r$-dimensional linear system, and $\bar{D}_{\lambda}$ is a $(d-r-1)$-plane in $\mathbb P^{g-1}$ spanned by $p_1(\lambda),...,p_d(\lambda)$.

Now apply to your case where $g=3$ and $C$ is hyperelliptic, and the linear system is a $g_1^2$ (which means $d=2$ and $r=1$), so the projective tangent cone of the isolated singularity at $W_2$ is just the canonical curve of $C$, i.e., a smooth conic in $\mathbb P^2$. Smoothness implies that the double point is nondegenerate, so it is an ordinary double point.