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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 …

Harris and Morrison point you (after stating the theorem in 1.5.4) to Clebsch's Zur Theorie der Rieman'schen Flachen Math Ann. 6 216-230, 1872. My German is not that good, but section 2 seems convinci …

There is the outrageously expensive "Geometry of the quintic" by Jerry Shurman, which discusses both Klein and Doyle-McMullen approaches (and then some more).

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 …

The "standard" definition of a K3 surface is field independent (unless you are a physicist): $p_g=1, q=0$, and trivial canonical class. Some results: Mumford and Bombieri showed that you get (just …

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 sequence stabilizes because any increasing sequence of bounded integers (the dimensions of the images of $X$) stabilizes, but I assume you mean something different. Suppose that $X\to|L|$ has alr …

The classical (pre Deligne-Mumford) approach is to map $\mathcal{M}_g$ into $\mathcal{A}_g$ using the Torreli map. Whereas the classical way to see that $\mathcal{A}_g$ is quasi-projective is to defin …

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 …