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I'm interested in the $K$-rational points on an imaginary hyperelliptic curve, where $K$ can be assumed to be a perfect field. I don't have much experience with blow-ups in algebraic geometry, so I would like to make sure that I'm not missing any $K$-rational points, besides the obvious ones.

From the reading I've done on hyperelliptic curves, I know that projective closure of equations of the form $y^2 + h(x) = f(x)$ over $K$, where $\deg(h) \geq g$ and $f$ is monic of degree $2g+1$, have a singularity at the point at infinity, and I know that taking the blow-up of this point yields one point at infinity (unlike the real case, where it yields two points).

However, none of the sources I've consulted mention what the side-effects are of blowing up this point. If we're not adding any points at infinity, are we instead changing the unique point at infinity somehow, or adding some affine points? What do these points look like? Are they defined over $K$, and if not, what does the extension they're defined over look like?

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The blow-up of the hyperelliptic curve is a special case of a general procedure called "resolution of singularities" of an algebraic curve, see here. In the case of this particular curve, it is a curve in $3$-dimensional space (not a plane curve) that is in one-to-one correspondence with the original curve, but the point corresponding to the point at infinity is now a smooth point, not a singularity. Because the hyperelliptic curve has a special form, it turns out that none of that is needed to construct and study the Jacobian variety of the curve; in fact, it is easier to do with the plane curve despite its singularity at infinity. If you have not done so already, check out the appendix on hyperelliptic curves over finite fields (of arbitrary characteristic) to Koblitz's Algebraic Aspects of Cryptography, see here.

  • A.J. Menezes, Y.H. Wu, and R. Zuccherato. An elementary introduction to hyperelliptic curves. N. Koblitz (Ed.), Algebraic Aspects of Cryptography, Springer, Berlin (1998), pp. 155–178.

The purpose of the appendix to the book is to show how to work with the Jacobian using only the plane curve, without having to worry about resolution of the singularity. That is why it is called "elementary".

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