41 votes
Accepted

Why not add cuspidal curves in the moduli space of stable curves?

If you add cuspidal curves, then $\overline{\mathcal{M}}_{1,1}$ will no longer be separated, which is the scheme/stack analogue of Hausdorff. Specifically, consider the families $$y_1^2 = x_1^3 + t^6 \...
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31 votes
Accepted

How did Riemann prove that the moduli space of compact Riemann surfaces of genus $g>1$ has dimension $3g-3$?

Riemann combines what is called Riemann-Roch and Riemann-Hurwitz nowadays. He considers the dimension of the space of holomorphic maps of degree $d$ from the Riemann surface of genus $g$ to the sphere....
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27 votes
Accepted

Polynomials with the same values set on the unit circle

This is a special case of the main theorem in the paper by I. N. Baker, J. A. Deddens, and J. L. Ullman, A theorem on entire functions with applications to Toeplitz operators, Duke Math. J. Volume 41, ...
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27 votes
Accepted

Motivation for zeta function of an algebraic variety

The definition using exponential of such an ad hoc looking series is admittedly not too illuminating. You mention that the series looks vaguely logarithmic, and that's true because of denominator $m$. ...
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  • 23.7k
23 votes
Accepted

Realizing algebraic curves as complete intersections

1) The genus of a complete intersection of multidegree $(d_1,\ldots ,d_{n-1})$ in $\mathbb{P}^n$ is $g=1+\frac{1}{2} d_1\cdot \ldots \cdot d_{n-1}(\sum d_i-n-1)$ (just compute the degree of the ...
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  • 34.5k
22 votes
Accepted

Rigorous version of heuristic argument for genus-degree formula?

Yes, this argument can be made rigorous. One needs three steps. Step 1. Show that there is at least one smooth plane curve of degree $d$ with the expected genus. Essentially, the proof is given by ...
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22 votes
Accepted

Is a one-dimensional compact complex analytic space necessarily projective?

Yes, every proper 1-dimensional complex-analytic space $X$ admits a closed immersion (in the sense of locally ringed spaces over $\mathbf{C}$) into an analytic projective space and more specifically ...
22 votes
Accepted

Modular forms from counting points on algebraic varieties over a finite field

The correct setting for this construction turns out to be projective varieties, so let me suppose we have a smooth variety $X$ inside $\mathbf{P}^N$, for some $N \ge 1$, defined by the vanishing of ...
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20 votes
Accepted

Is it a new discovery on conic section?

It suffices to consider the case when $\Omega$ is a circumcircle, so let it be. At first, the points $A_b, A_c, B_c, B_a, C_a, C_b$ lie on a conic if and only if $$ \frac{AB_a\cdot AB_c}{AC_a\cdot ...
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  • 88.5k
19 votes
Accepted

Why is the section conjecture important?

You can start here "Fermat's last theorem" and anabelian geometry?? In particular, I mention there that: At some point Deligne thought he had a proof that the section conjecture implied ...
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18 votes
Accepted

History of the connection between Riemann surfaces and complex algebraic curves

The connection came from the paper by Dedekind and Weber "Theorie der algebraischen Functionen einer Veranderlichen", Crelle's Journal, 1882. In this paper the authors recover the theory by Riemann (...
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  • 8,201
17 votes
Accepted

Are there integer solutions to $3y^2 = 4x^3-1$ other than $(1,1)$ and $(1,-1)$?

The projective form of your curve is $3y^{2} z = 4x^{3} - z^{3}$. This has three obvious points: $(1 : 1 : 1)$, $(1 : -1 : 1)$, and $(0 : 1 : 0)$. Your curve is isomorphic over $\mathbb{Q}$ to the ...
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  • 17.7k
17 votes
Accepted

History of Study's Lemma?

I found the lemma on page 63 of Study's Einleitung in die Theorie der Invarianten linearer Transformationen auf Grund der Vektorenrechnung (1923). The source cited for the proof is page 202 of Study'...
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16 votes

History of the connection between Riemann surfaces and complex algebraic curves

I have always felt this was due to Riemann himself: especially in: Theory of Abelian Functions, 1857. Of course the association of a Riemann surface to an algebraic curve is generally attributed to ...
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  • 11.5k
16 votes

Automorphisms of cartesian products of curves

That is certainly not true. Consider the case that $C$ is an elliptic curve. Then $\text{Aut}(C\times C)$ contains $\text{GL}(2,\mathbb{Z})$ as a subgroup.
16 votes

Motivations to study the cohomology of the moduli space of curves

As the title to Mumford's famous paper "Toward an enumerative geometry..." suggests, knowing the cohomology / cycle theory of the moduli space of curves allows one to answer enumerative geometry ...
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  • 5,784
15 votes

Is Proposition 2.6 in J. Silverman's book Arithmetic of Elliptic Curves correct?

The reason is that there are two ways of thinking about "points". Let $A$ be a ring. Then, define: A scheme-theoretic/topological point of Spec $A$ is a prime ideal of $A$. A geometric/functorial ...
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  • 9,055
15 votes

How did Riemann prove that the moduli space of compact Riemann surfaces of genus $g>1$ has dimension $3g-3$?

The original paper of Riemann is his celebrated "Theorie der Abel'schen Functionen" in Crelle's Journal of 1854. This paper can be found online at https://www.maths.tcd.ie/pub/HistMath/People/Riemann/...
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  • 1,178
15 votes

Polynomial values are powers of two

I'll prove a stronger statement. Let $S$ be a finite set of primes. I claim there is a $c_{n,S}$ such that a polynomial $f$ with rational coefficients cannot take only values that are $S$-units on $\{...
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  • 116k
14 votes
Accepted

Quotients of curves of genus $4$ by a free $\mathbb{Z}/ 3 \mathbb{Z}$-action

Yes. Start from a genus 2 curve $C_2$, and choose a point of order 3 in $JC_2$, giving rise to an étale $\mathbb{Z}/3$-covering $C_4\rightarrow C_2$. Then $C_4$ cannot be hyperelliptic: a $g^1_2$ on ...
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  • 34.5k
14 votes
Accepted

Do $\mathbb{A}^1-S$ and $\mathbb{A}^1-\{0,1\}$ have a finite etale cover in common?

The answer is positive if and only if $\mathbb{A}^1\setminus S$ is an arithmetic curve, i.e., $\pi_1(\mathbb{A}^1\setminus S)\subset \mathrm{Aut}(\mathbb{H}) = PSL_2(\mathbb{R})$ is an arithmetic ...
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13 votes

what is the maximum number of rational points of a curve of genus 2 over the rationals

Here is some more information. The curve that establishes the current record is obtained from a K3 surface $S$ that was found by Noam Elkies. $S$ is a double cover of ${\mathbb P}^2$ ramified above a ...
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13 votes
Accepted

Original reference for Riemann's inequality

B. Riemann, Theorie der Abel'schen Functionen, Journal für die reine und angewandte Mathematik 54, 101–155 (1857). Here is a description of this contribution, by Jeremy Gray: In this 1857 paper ...
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13 votes

Motivation for zeta function of an algebraic variety

Exercise 4.8 of Enumerative Combinatorics, vol. 1, second ed., and Exercise 5.2(b) in volume 2 give an explanation of sorts for general varieties over finite fields. According to Exercise 4.8, a ...
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13 votes
Accepted

Examples of plane algebraic curves

How about the affine plane curves $\Phi_n(c,t)=0$ that classify $(c,t)$ such that $t$ is a point of exact period $n$ under iteration of the quadratic map $f_c(X)=X^2+c$? These are often called ...
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13 votes
Accepted

Curve with no embedding in a toric surface

A generic curve of genus $5$ is not a hypersurface in a toric surface. This argument is going to use conceptual ideas from Haase and Schicho's paper "Lattice polygons and the number $2i+7$", ...
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13 votes
Accepted

Visualizing genus-two Riemann surfaces: from the three-fold branched cover to the sphere with two handles

The picture (produced by Nick Schmitt) of the Lawson surface of genus 2 might help: It shows the genus 2 Riemann surface given by the algebraic equation $$y^3=\frac{z^2-1}{z^2+1}.$$ The lines show ...
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  • 6,325
13 votes

Polynomial values are powers of two

Yes, such $c_n$ is bounded by something effective. Below is a cubic bound, which probably may be improved. (Update: see $n^2\log n$ upper bound by Will in the comments.) Assume that $f(x)$ is a power ...
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  • 88.5k
12 votes
Accepted

Natural model for genus $6$ curves

On the canonical model $C_{10} \subset \mathbb{P}^5$ of a smooth curve of genus six there exist five special $g_4^1$, obtained as follows: we take an arbitrary point on the curve, and the remaining ...
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12 votes

Priority for lemniscate of Gerono?

The lemniscate $x^4-x^2+y^2=0$ was discussed in Gerono's Géométrie Analytique from 1854, see screenshot, while Lissajous's "Mémoire sur l'étude optique des mouvements vibratoires" is from 1857. The ...
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