### ag.algebraic geometry - Is every curve birational to a smooth affine plane curve?

Is every curve over $\mathbf{C}$ birational to a smooth affine plane curve?Bonnie Huggins asked me this question back in 2003, but neither I nor the few people I passed it on to were able to answer it. It is true at least up to genus 5....Read more

### algebraic curves - Determining numerical class of divisors inside Jacobian

Let $C$ be a smooth projective curve of genus $g$, let $c$ be a point in $C$. Let $(n_1,\dots, n_{g-1})$ be a $(g-1)$ tuple of nonzero integers. Consider the image $f_{(n_i)}\colon C^{g-1}\to \mathrm{Pic}^0(C)$ given by $$(x_i)\mapsto \mathcal{O}_C\left(\sum n_i(x_i-c)\right),$$the image $\mathrm{Im}(f_{(n_i)})$ is a divisor in $\mathrm{Pic}^0(C)$. If $C$ is general, then $NS(C)$ is generated by theta divisor $\Theta$, so we can write numerical classes as a multiple of the theta divisor,$$\mathrm{Im}(f_{(n_i)})=d(n_1,\dots,n_{g-1}）\Theta.$$Is ...Read more

### ag.algebraic geometry - Families of curves for which the Belyi degree can be easily bounded

I know (edit: three) families of smooth projective connected curves over $\bar{\mathbf{Q}}$ for which the Belyi degree is not hard to bound from above. The modular curves $X(n)$. They are constructed by compactifying the quotient $Y(n) = \Gamma(n)\backslash \mathbf{H}$. The natural morphism $X(n) \longrightarrow X(1)$ is Belyi of degree $n^2$ (up to a constant factor). This also bounds the Belyi degree of a modular curve given by a congruence subgroup $\Gamma$. In general, Zograf proves that the Belyi degree of a (classical congruence) modular ...Read more

### moduli spaces - Trigonal curves of genus three: can their Galois closure be non-abelian

Let $X$ be a curve of genus three which is not hyperelliptic. Then $X$ is trigonal, i.e., there exists a finite morphism $X \to \mathbf P^1$ of degree $3$.Let $Y\to X \to \mathbf P^1$ be a Galois closure of a trigonal curve. Then, unless $X\to \mathbf P^1$ is the Klein curve, the map $Y\to \mathbf P^1$ is of degree $6$. Now, surely the latter map could have non-abelian Galois group, but I don't know any explicit examples. Can somebody give me an explicit example for which it is clear that the Galois group over $\mathbf P^1$ of the Galois closur...Read more