10 votes
Accepted

Definition of modular curve associated to $\Gamma(N)$

This is a subtle issue (which has come up before on this site several times, see e.g. is the modular curve X(N) defined over Q? for a related question). Your $S(N)$ is naturally a scheme over $\mathbb{...
David Loeffler's user avatar
10 votes

Applications of derived categories to "Traditional Algebraic Geometry"

The global Torelli Theorem for cubic fourfolds says the following. Let $X_1 \subset \mathbb{P}^5$ and $X_2 \subset \mathbb{P}^5$ be smooth cubic fourfolds. The fourfolds $X_1$ and $X_2$ are ...
6 votes

Conformal map between flat and hyperbolic torus with a boundary

Finding explicit relations between flat and hyperbolic structures is notoriously difficult. We struggled with this in the case of Dyck's surface in the Transactions article http://doi.org/10.1090/...
Mikhail Katz's user avatar
  • 15.1k
5 votes
Accepted

Is multiplication by $d$ on the Jacobian of a nodal curve étale?

Lemma. Let $G$ be a commutative group scheme of finite type over a field $k$, and let $d$ be a positive integer invertible in $k$. Then the multiplication by $d$ map $[d] \colon G \to G$ is finite ...
R. van Dobben de Bruyn's user avatar
3 votes
Accepted

Examples when algebraic 1-stack = derived enhancement?

If $S$ is a smooth projective variety of dimension $d$, then the derived stack $X=Bun_G(S)$ has cotangent complex perfect of amplitude $[-(d-1), 1]$. If $S=C$ is a curve then it is in $[0,1]$, in ...
crystalline's user avatar
3 votes

Finer classification of semistable sheaves

Sometimes you can perturb polarization (or more generally, stability condition) so that the slopes of $E_1$ and $E_2$ become different. Then the moduli space for the slightly perturbed stability is ...
Sasha's user avatar
  • 37k
3 votes

inclusion of moduli spaces induced by morphism between certain universal families

The moduli space of stable sheaves is by definition a scheme that corepresents the functor of families of stable sheaves. In particular, for any family of stable sheaves there is a canonical morphism ...
Sasha's user avatar
  • 37k
3 votes
Accepted

Is there a *relative* moduli stack of objects functor?

Let me give the relative construction. We'll say our geometric objects are presheaves on some category $Aff$, and we'll denote sheaves of categories by $2QCoh(-)$. There's a functor $$P(Aff)_{/B}^{op} ...
Mori B.'s user avatar
  • 68
2 votes

Moduli of smooth curves

Is Arbarello's 74 paper (specifically, theorem 3.27 there) Weierstrass points and moduli of curves an easy enough argument ?
David Lehavi's user avatar
  • 4,299
2 votes

Moduli of smooth curves

Seeing that there are already two answers that are not entirely convincing, I decided to copy my comment to an answer. If one assumes the classical result that, for genus g≥2, the moduli space of ...
Mikhail Katz's user avatar
  • 15.1k
2 votes
Accepted

When is the morphism from the Hilbert scheme to the moduli scheme of stable sheaves an isomorphism?

This works for any smooth projective variety $X$ under the assumption $$ \mathrm{Pic}^0(X) = 0 $$ and any $Z$ of codimension at least 2. For the proof see Lemma B.5.6 in Kuznetsov, Alexander G.; ...
Sasha's user avatar
  • 37k
2 votes

One question about K-moduli space of smooth plane conic curves

There are two ways to see that $X$ is isomorphic to $\mathbb{P}(1,1,4)$, both using the fact that $X$ isotrivially degenerates to $\mathbb{P}(1,1,4)$. (1) By Hacking's paper https://arxiv.org/pdf/math/...
Yuchen Liu's user avatar
  • 1,038
2 votes

Symmetric differential forms on moduli space of curves

This is not exactly what you are asking, but if you allow the symmetric differentials to have logarithmic poles along the boundary of the Deligne-Mumford compactification, the answer is "yes"...
Daniel Litt's user avatar
  • 22.2k
1 vote
Accepted

Moduli of smooth curves

Here is yet another approach. If $M_g$ were proper, its image in $A_g$ under the Torelli map would be closed. But products of lower dimensional Jacobians are in the closure. This last result is ...
Samir Canning's user avatar
1 vote

Isomorphism between $\operatorname{End}_0(E)$ and $\operatorname{End}_0(E')$ as Lie algebra bundles

With this type of question, it is better to write to one of the authors directly. $\DeclareMathOperator\Fl{Fl}\DeclareMathOperator\End{End}$First, a vertical vector field is given by a global vector ...
Vicente Muñoz's user avatar

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