# Questions tagged [moduli-space]

A Moduli space is a space in algebraic geometry whose points are geometric objects or isomorphism classes of these kinds of objects.

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### Computing the virtual dimension of the moduli space $M(L, \beta)$

In Auroux's paper "MIRROR SYMMETRY AND T-DUALITY IN THE COMPLEMENT OF AN ANTICANONICAL DIVISOR", in section §3 p.7 the author presents the moduli space $M(L, β)$ of J-holomorphic discs with ...
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### Construction of Deligne-Mumford curve moduli space

The standard construction of $\newcommand{\PP}{\mathbb{P}} \DeclareMathOperator{\Hilb}{Hilb} \DeclareMathOperator{\PGL}{PGL} M_g$ is as follows (At least for stable curves and other technical ...
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### Morphism of family of varieties determined on rational points

I am currently studying various Moduli problems and in order to check whether some families have non-trivial automorphisms, I have the strong intuition that the following should hold: Let $k$ be an ...
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### representing compactified stack as a global quotient

Let $M$ be a DM stack over a field $k$. Assume I managed to represent it as $[X/G]$ for some scheme $X$ and a group scheme $G$. Does that imply that the DM compactification $\overline{M}$ of $M$ is ...
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### Rational points on the the reduction mod $p$ of $X_0(N)$

Consider the modular curve $X_0(N)$ over $\mathbb Q$ and for $p\mid N$ consider the reduction modulo $p$ of $X_0(N)$. Let's denote this curve with the symbol $X_p$ (we know that it is a singular curve ...
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### Degree of Hodge bundle

Let $H$ be the Hilbert scheme parametrizes subschemes of $\mathbb P^{5g-6}$ with Hilbert polynomial $p(m)=(6g-6)m+(1-g)$ (for example, curves with genus $g$ embedding by the canonical bundle to the ...
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### What does "intersect properly" mean?

A corollary in The Geometry of Moduli Space of Sheaves (Huybrechts, Lehn) says: Let $X$ be a normal closed subscheme in $P^{N}$ and $k$ an infinite field. Then there is a dense open subset $U$ of ...
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### Why is the definition of the dual sheaf is independent of the ambient space?

I'm reading "The Geometry of Moduli Space of Sheaves" (Huybrechts,Lehn). He gives a new definition of the dual sheaf: $\$Let E be a coherent sheaf of dimension $d$ ,and let $c=n-d$ be its ...
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### Learning about Moduli spaces of sheaves

I am a Ph.D. student and starting a side project with a fellow student on Moduli spaces. Our plan was to start with the book on Invariants and Moduli by Mukai (starting from chapter 5) and use the ...
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### Relationship between GIT and coarse moduli spaces

I'm trying to understand how a generic algebraic geometer constructs coarse moduli spaces. I'm familiar with the definition, and how it is usually quite involved to show that a space has the coarse ...
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