# Requirements on holomorphic embedding in terms of the associated line bundle

Consider a holomorphic embedding $f$ of a genus $g$ Riemann surface $\Sigma_g$ into a generic quintic $Q \subset \mathbb{CP}^4$

This is the same as giving the (very ample) line bundle over the curve $\Sigma_g$

$$L = f^* \mathcal{O}_{\mathbb{CP}^4}(5)$$

(or should it be $\mathcal{O}(1)$? does this suffice to guarantee the map is to $Q$?)

We can endow both the Riemann surface and $\mathbb{CP}^4$ with antiholomorphic involutions, call them respectively $\Omega$ and $\sigma:(x_1:x_2:x_3:x_4:x_5)\mapsto (\bar x_2:\bar x_1:\bar x_4:\bar x_3:\bar x_5)$ and require $f$ to be equivariant.

(Define also $\mathcal{L}$ as the pointwise fixed locus of $\sigma$ and specify a degree of the map $d=f_*([\Sigma_g])\in H_2(X,\mathcal{L};\mathbb{Z})$.)

QUESTION 1: How is the equivariance requirement translated in terms of line bundles?

Moreover, we can imagine $f$ to develop a node on top of $L$, and then smooth it in different ways.

QUESTION 2: What are the effects of the smoothings in terms of line bundles?

• I see from your new question that I misunderstood your comment in your previous question. I thought you were talking about a curve of degree $5$ in $\mathbb P^4$, but now it seems $Q$ is a degree $5$ hypersurface. – Brenin Mar 19 '14 at 11:52
• @Brenin maybe I was not precise: $Q$ is the vanishing locus of a degree 5 polynomial in $\mathbb{P}^4,$ and thus it is a hypersurface, as you say; if I pull back the line bundle over $\mathbb{P}^4$ with sections some degree 5 polynomials I get a line bundle over the curve or Riemann surface, which as you explained previously is the same (up to isomorphism) as the embedding $f$: do you think this is correct? – jj_p Mar 19 '14 at 12:05
• Yes, I guess miraculously my misunderstanding did not cause problems. – Brenin Mar 19 '14 at 13:57
• @Brenin Do you think that to have an embedding into the quintic it's enough to pull back $\mathcal{O}(5)$ instead of $\mathcal{O}(1)$? – jj_p Mar 19 '14 at 14:59