Search Results

For an image summarizing the Richelot-Humbert genus 2 AGM construction see e.g. page 22 of https://arxiv.org/pdf/1006.3408 Humbert: http://www.numdam.org/item/JMPA_1901_5_7__395_0.pdf Richelot: http …

For non hyper-elliptic curves and $d < g$ you can use a geometric Riemann-Roch argument: A map $f:C\to\mathbb{P}^1$ of degree $d<g$ is defined by choice of a codimension $2$ linear projective subspace …

Is Arbarello's 74 paper (specifically, theorem 3.27 there) Weierstrass points and moduli of curves an easy enough argument ?

I believe I have an example in genus 5: Humbert curves (see either Varley's "Weddle's Surfaces, Humbert's Curves, and a Certain 4-Dimensional Abelian Variety", or exercise batch F in chapter 6 of ACGH …

Moduli theory of quartic Del Pezzp surfaces goes back to Coble. I hope the following tow refrences are usefull: moduli of quartic del pezzo surfaces (Colombo, van Geemen, Looijenga) on the moduli of d …

AFAIK, these are know only up to genus 3. Genus 2: Igusa (classical). Hyperelliptic genus 3: Shioda (classical). Non hyperelliptic genus 3: a decade ago by Dixmier & Ohno - see https://www.win.tue.n …

No, there is no such $S$: EDIT: (BIG) GAP BELOW I compute limits of certain linear series in the Hurwitz scheme, and then I make claims about limits of other (bigger) linear systems taken over curves …

I'd speculate that these are the linear systems on the Kummer surface given by twice the curves described in Hudson's "Kummer's quartic surface" sections 90 and 91: identifying the Kummer surface with …

Let $C$ be a non hyperelliptic complex algebraic curve of genus $g$, then the vector space $I_2(C)$ of quadrics containing the canonical image of $C$ is $\binom{g+1}{2}-h^0(2\omega_C) = (g-2)(g-3)/2$ …

The phase space of Kovalevskaya's top is an Abelian surface. If you fix some natural invariant you usually get a curve of small genus. If memory serves right, fixing the energy gives you a genus 2 cur …

As Robin and Fedor observed the variety in question is a quartic Del Pezzo surface. There is a nice treatment in Igor Dolgachevs "Topics in classical algebraic geometry I" section 8.5 (including expli …

Consider some $a,b,c\in E$. Then $a\oplus b\oplus c=0_E$ in the group $E$ iff $a+b+c = 3\cdot 0_E$ in $\mathrm{Pic}^3(E)$ iff $a,b,c$ are colinear in the complete linear system $|\mathcal{O}_E(3\cdot …

There is a nice computational perspective in Bayer and Mumford's What Can Be Computed in Algebraic Geometry? pages 4,5.

With 2^2 isogeny it never works: F. Richelot, De transformatione integralium Abelianorum primi ordinis comentatio. J. reine angew. Math. 16 (1837) 221-341 G. Humbert, Sur la transformation ordinaire …

In Alain Roberts "Elliptic curves: notes from postgraduate lectures given in Lausanne 1971/72" page 11 (available on google books unless you already tried to read another chapter), there is a hand dra …