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
$$y_1^2 = x_1^3 + t^6 \...
- 150k
32
votes
Accepted
How did Riemann prove that the moduli space of compact Riemann surfaces of genus $g>1$ has dimension $3g-3$?
Riemann combines what is called Riemann-Roch and Riemann-Hurwitz nowadays.
He considers the dimension of the space of holomorphic maps of degree $d$ from the Riemann surface of genus $g$ to the sphere....
- 87.4k
27
votes
Accepted
Polynomials with the same values set on the unit circle
This is a special case of the main theorem in the paper by
I. N. Baker, J. A. Deddens, and J. L. Ullman,
A theorem on entire functions with applications to Toeplitz operators,
Duke Math. J.
Volume 41, ...
- 87.4k
27
votes
Accepted
Motivation for zeta function of an algebraic variety
The definition using exponential of such an ad hoc looking series is admittedly not too illuminating. You mention that the series looks vaguely logarithmic, and that's true because of denominator $m$. ...
- 26.3k
23
votes
Accepted
Is a one-dimensional compact complex analytic space necessarily projective?
Yes, every proper 1-dimensional complex-analytic space $X$ admits a closed immersion (in the sense of locally ringed spaces over $\mathbf{C}$) into an analytic projective space and more specifically ...
22
votes
Accepted
Modular forms from counting points on algebraic varieties over a finite field
The correct setting for this construction turns out to be projective varieties, so let me suppose we have a smooth variety $X$ inside $\mathbf{P}^N$, for some $N \ge 1$, defined by the vanishing of ...
- 35.8k
20
votes
Accepted
Is it a new discovery on conic section?
It suffices to consider the case when $\Omega$ is a circumcircle, so let it be.
At first, the points $A_b, A_c, B_c, B_a, C_a, C_b$ lie on a conic if and only if
$$
\frac{AB_a\cdot AB_c}{AC_a\cdot ...
- 99k
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 ...
- 19.9k
17
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 ...
- 5,890
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'...
- 173k
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.
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 ...
- 9,918
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/...
- 1,364
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 $\{...
- 131k
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 ...
- 36.9k
14
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 ...
- 173k
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 ...
- 9,387
14
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 ...
- 45.3k
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 ...
- 11.2k
13
votes
Accepted
Faltings theorem and number of singularities
The definition of the geometric genus in terms of (d-1)(d-2)/2 minus the contributions of the singularities is not a great one. It's better to give a more intrinsic definition, as the dimension of the ...
- 131k
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 ...
- 48.7k
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 $2i+7$", ...
- 150k
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 ...
- 6,615
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 ...
- 99k
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 ...
- 64.8k
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 ...
- 173k
12
votes
Smoothen a nodal curve
This is correct that you can always "deform" a nodal curve into a smooth one. A good reference for this fact is Corollary $7.11$ in these notes by Talpo and Vistoli.
UPD: As Qixiao points out Talpo ...
- 2,841
12
votes
Accepted
Weak Mordell-Weil for EC using Chevalley-Weil theorem
The multiplication-by-$m$ map $[m]:E\to E$ is unramified, so there exists a finite set of primes $S$, depending only on $E$ and $m$, so that for every $P\in E(K)$, the field generated by the ...
- 45.3k
11
votes
Accepted
Given a curve $C$, does there exist a rational function on $C$ totally ramified at two given points?
The answer is in general no. More precisely, the following holds.
A finite map $f \colon C \to \mathbb P^1$ as in the question exists if and only if $\mathcal{O}_C(x-y)$ is a point of finite order ...
- 64.8k
11
votes
Accepted
Shafarevich conjecture for abelian varieties
Let $B$ be a smooth projective curve over an algebraically closed field of characteristic zero. Let $K$ be the function field of $B$. Let $S$ be a finite set of closed points of $B$.
You might find ...
- 9,387
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