# Tagged Questions

An algebraic curve or plane algebraic curve is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables.

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### Polynomials with the same values set on the unit circle

Assume that $P(z)$, $Q(z)$ are complex polynomials such that $P(S)=Q(S)$, where $S=\{z\colon |z|=1\}$ (equality is understood in the sense of sets, but I do not know the answer even for multisets). ...
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### Curves which are not covers of P^1 with four branch points

The following interesting question came up in a discussion I was having with Alex Wright. Suppose given a branched cover C -> P^1 with four branch points. It's not hard to see that the field of ...
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### Which 'well-known' algebraic geometric results do not hold in characteristic 2?

A smooth curve $X$ in $\mathbb{P}^n$ is strange if there is a point $p$ which lies on all the tangent lines of $X$. Examples are $\mathbb{P}^1$ is strange and so is $y=x^2$ in characteristic $2$. ...
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### Mumford conjecture: Heuristic reasons? Generalizations? … Algebraic geometry approaches?

The Mumford conjecture states that for each integer $n$, we have: the map $\mathbb{Q}[x_1,x_2,\dots] \to H^\ast(M_g ; \mathbb{Q})$ sending $x_i$ to the kappa class $\kappa_i$, is an isomorphism in ...
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### The work of E. Artin and F. K. Schmidt on (what are now called) the Weil conjectures.

I was reading Dieudonne's "On the history of the Weil conjectures" and found two things that surprised me. Dieudonne makes some assertions about the work of Artin and Schmidt which are no doubt ...
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### Do all curves have Néron models

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$. Does there exist a Néron model $\mathcal X$ for $X$ over $O_K$? By a Néron model, I mean ...
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### 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 ...
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### Links between Riemann surfaces and algebraic geometry

I'm taking introductory courses in both Riemann surfaces and algebraic geometry this term. I was surprised to hear that any compact Riemann surface is a projective variety. Apparently deeper links ...
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### A historical question: Hurwitz, Luroth, Clebsch, and the connectedness of $\mathcal{M}_g$

The connectedness of the moduli space $\mathcal{M}_g$ of complex algebraic curves of genus $g$ can be proven by showing that it is dominated by a Hurwitz space of simply branched d-fold covers of the ...
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### Does there exist a curve of degree 11 having 15 triple points?

Does there exist an irreducible curve of degree 11 in the projective plane which would have 15 triple points? For information, such a curve would be rational, if it exists, and would be smooth at ...
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### Which curves can be found on Abelian varieties ?

We know tha each genus 2 curve is embedded into its degree 1 Jacobian. Under which conditions on $C$, $A$, $g$ and $n$ is it possible for a genus $g$ smooth curve $C$ to be embedded in an Abelian ...
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### Tame morphism from a curve to $\mathbb{P}^1$

Let $k$ be an algebraically closed field of characteristic $p\ge 0$. Let $C$ be a smooth projective curve over $k$. Is it possible to find a map $C \to \mathbb{P}^1$ that is tamely ramified at every ...
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### Do mapping classes have gonality?

(This question was discussed by people at the PCMI workshop on moduli spaces, without any clear resolution, so I thought I'd throw it open to MO.) The hyperelliptic mapping class group is (by ...
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### A good example of a curve for geometric Langlands

I'm currently working through Frenkel's beautiful paper: http://arxiv.org/PS_cache/hep-th/pdf/0512/0512172v1.pdf. I'm looking for a good example of a projective curve to get my hands dirty, and go ...
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### History of the connection between Riemann surfaces and complex algebraic curves

As noted in the question "Links between Riemann surfaces and algebraic geometry", there are strong connections between Riemann surfaces and algebraic geometry - for example, compact Riemann surfaces ...
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### Does every hyperbolic curve over a finite field have an etale cover with a real Frobenius eigenvalue?

More precisely: let X/F_q be a smooth projective algebraic curve of genus at least 2. Does there always exist a curve Y/F_{q^d} with a finite etale projection Y -> X, such that one of the Frobenius ...
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### Riemann surfaces: explicit algebraic equations

Suppose $\Gamma$ is a nice discrete subgroup of $SL(2,\mathbb{R})$ such that the genus of the Riemann surface $\mathbb{H}/\Gamma$ is larger than 1. We know that this Riemann surface is also an ...
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### New(?) reciprocity law

Consider three functions $f, g$ and $h$ on a smooth curve $X$ over $\mathbb{C}$. I have found the following equality: $$\sum (res(f\frac{dg}{g})\frac{dh}{h}-res(f\frac{dh}{h})\frac{dg}{g})=0.$$ Here ...
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### Does the moduli space of genus three curves contain a complete genus two curve

Inspired by the question Does the moduli space of smooth curves of genus g contain an elliptic curve and its amazing answers, I ask (pure out of curiosity) whether the moduli space $M_3$ of (smooth ...
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### When does the Torelli Theorem hold?

The Torelli theorem states that the map $\mathcal{M}_g(\mathbb{C})\to \mathcal{A}_g(\mathbb{C})$ taking a curve to its Jacobian is injective. I've seen a couple of proofs, but all seem to rely on the ...
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### Vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$

I have a question about vector bundles on the algebraic surface $\mathbb{P}^1\times\mathbb{P}^1$. My motivation is the splitting theorem of Grothendieck, which says that every algebraic vector bundle ...
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### What is the Euler characteristic of a Hilbert scheme of points of a singular algebraic curve?

Let $X$ be a smooth surface of genus $g$ and $S^nX$ its n-symmetrical product (that is, the quotient of $X \times ... \times X$ by the symmetric group $S_n$). There is a well known, cool formula ...
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### Curves with negative self intersection in the product of two curves

I wonder if the following is known: Are there two compact curves C1 and C2 of genus>1 defined over complex numbers, such that their product contains infinite number of irreducible curves of negative ...
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### Are curves with fractional points' uniquely determined by their residual gerbes?

One makes precise the vague notion of "curve with a fractional point removed" (see for instance these slides) using stacks -- one should really consider Deligne-Mumford stacks whose coarse spaces are ...
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### Why is a general curve automorphism-free?

Fix an algebraically closed field $k$. Why is the general curve over $k$ of genus $g \ge 3$ automorphism-free? I am particularly interested in seeing an argument that does not go by induction and ...
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### Intuition for thinking about $R$-module of Kähler differentials, universal receptacles, derivations?

Suppose $k$ is a field of characteristic zero, and $R$ is a $k$-algebra. The $R$-module of Kähler differentials $\Omega_{R/k}$ of $R$ over $k$ with generators $\{dr\}_{r \in R}$ is the module subject ...
Question: Is it true that for every smooth compact complex curve $C$ there exists a smooth curve $C'$ in $\mathbb CP^2$ that admits a non-trivial morphism (i.e. holomorphic map) $C'\to C$? Motivation....