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Let $X$ be a curve of genus at least two. Then one can associate an abelian variety to $X$; this is the Jacobian.

Let $X\to Y$ be a double cover of curves. Then we can associate an abelian variety to $X\to Y$; this is the Prym variety.

To a K3 surface one can associate an abelian variety; this is the Kuga-Satake variety.

Question. Can we associate a K3 surface to a curve (or just some special class of curves) in a reasonable/useful way?

Question 2. Can we do the same with $X\to Y$ covers of degree $d$ (or just degree two)?

This question is vague. I understand. I'm really just asking whether there is some way to go from a curve of genus at least two to a K3 surface.

Elliptic curves are abelian varieties. So for a genus one curve $E$ over $\mathbf C$ one could ask whether there is a K3 surface associated to it. I think in this case you could maybe write down an elliptic K3 surface $\mathcal E\to \mathbf P^1$ whose generic fibre is $E$. Still need to figure out if this is useful though...

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