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for questions on one dimensional algebraic varieties over any field, including questions of moduli, and questions about specific curves.

4 votes

etale cover of a hyperelliptic curve

Since nobody has answered this yet, here is my attempt, however Im not sure if it is exactly what you want. Hopefully it will at least lead you to a solution of your problem if you have not seen this …
Daniel Loughran's user avatar
4 votes
Accepted

Computing $H^1$ with coefficients in a torsion-free abelian group

I will focus attention on smooth projective varieties $X$ over $k$ with $\mathrm{Pic}(X_{\bar{k}})$ a free finitely generated abelian group, as they illustrate all the essential behaviour relevant to …
Daniel Loughran's user avatar
7 votes

Injectivity under flat base change of the Picard group on smooth projective curves

Yes this is true. It can be proved using the Hochschild-Serre spectral sequence plus Hilbert's theorem 90 (it is true more generally for any geometrically connected projective variety $X$). The Hochs …
Daniel Loughran's user avatar
4 votes
Accepted

Does a smooth relative curve $X/S$ embed into $\mathbb{P}^3_S$?

This answer addresses the second question: "If we assume the fibers of $\pi$ are curves of genus $0$, can we embed $X$ into $\mathbb{P}^2_S$?" The answer to this is also no (providing there is a singu …
Daniel Loughran's user avatar
9 votes

Obstruction and rational points on curves

I consider only smooth curves for simplicity. In which case this is expected to be true, but certainly not known in general. In fact, it is even expected that the Brauer-Manin obstruction is already e …
Daniel Loughran's user avatar
2 votes

Naive question on the Jacobian of a curve

If the Neron-Severi group of an abelian variety is $\mathbb{Z}$, then a principal polarisation, if it exists, is unique. This is the generic case, as well as for generic jacobians. To find counter-exa …
Daniel Loughran's user avatar
3 votes

Space of rational conics

Let $$X: \quad a_{0,0}x^2 + a_{1,1}y^2 + a_{2,2}z^2 + a_{0,1}xy + a_{0,2}xz + a_{1,2}yz = 0 \quad \subset \mathbb{P}^2 \times \mathbb{P}^5$$ be the total space of the family of all plane conics. Then …
Daniel Loughran's user avatar