The projective space $\mathbb{P}^n_S$ represents the functor $T \mapsto \left\lbrace \text{line bundle quotients } \mathcal{O}_T^{n+1} \twoheadrightarrow \mathcal{L} \right\rbrace / \cong$ The universal family is the tautological quotient $\mathcal{O}_{\mathbb{P}^n}^{n+1} \twoheadrightarrow \mathcal{O}(1)$.

The Grassmannian $\mathrm{Gr}(k, n)$ is a fine moduli space for the functor $T \mapsto \left\lbrace \text{locally free quotients } \mathcal{O}_T^n \twoheadrightarrow \mathcal{E}, \; \mathrm{rank}(\mathcal{E}) = k \right\rbrace / \cong$ The universal family is the tautological quotient bundle of rank $k$ on $\mathrm{Gr}(k,n)$.

The affine space $\mathbb{A}^{n+1}$ is a fine moduli space for monic polynomials of degree $n$: $T \mapsto \left\lbrace f(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_0 \;\middle|\; a_i \in \mathcal{O}(T) \right\rbrace$ The universal polynomial is $x^n + t_{n-1}x^{n-1} + \cdots + t_0$ over $\mathbb{A}^n = \mathrm{Spec}\, k[t_0, \ldots, t_{n-1}]$.

Consider the moduli functor $\mathcal{M}_{1,1}$ of elliptic curves (genus 1 curves with a marked point). Every elliptic curve $E$ has the automorphism $[-1]: E \to E$ (the inverse map), so $|\mathrm{Aut}(E)| \geq 2$. This functor is not representable by a scheme.

More precisely, if $\mathcal{M}_{1,1}$ were represented by a scheme $M$, then for any field $k$, the set of $k$-points $M(k)$ would be in bijection with isomorphism classes of elliptic curves over $k$. But one can construct a family $E \to \mathbb{P}^1 \setminus \lbrace 0,1,\infty \rbrace$ that is locally trivial in the etale topology but not globally trivial, contradicting representability.

Let $C$ be a smooth projective curve. The functor $T \mapsto \left\lbrace \text{line bundles on } C \times T \right\rbrace / \cong$ is not representable because every line bundle $\mathcal{L}$ has $\mathrm{Aut}(\mathcal{L}) = \mathbb{G}_m$. However, if we rigidify by fixing a point $p \in C$ and requiring a trivialization along $\lbrace p \rbrace \times T$, we obtain the representable Picard scheme $\mathrm{Pic}_C$.

The affine line $\mathbb{A}^1_j = \mathrm{Spec}\, \mathbb{Z}[j]$ is a coarse moduli space for elliptic curves over algebraically closed fields. The $j$-invariant provides the natural transformation. However, this is not a fine moduli space: there is no universal elliptic curve over $\mathbb{A}^1_j$.

Over a non-algebraically closed field $k$, there can exist non-isomorphic elliptic curves with the same $j$-invariant (twists), demonstrating the failure of the fine moduli property.

The coarse moduli space $M_g$ of smooth curves of genus $g \geq 2$ exists as a quasi-projective variety of dimension $3g - 3$. However, it is not a fine moduli space because:

• Curves can have nontrivial automorphisms (e.g., hyperelliptic involutions).
• The hyperelliptic locus in $M_g$ parametrizes curves with $\mathrm{Aut}(C) \supseteq \mathbb{Z}/2\mathbb{Z}$.

For $n \geq 3$, the moduli functor $\mathcal{M}_{1,1}[n]$ of elliptic curves with full level-$n$ structure is representable by a smooth affine curve $Y(n)$ over $\mathbb{Z}[1/n]$. The key point: a level-$n$ structure with $n \geq 3$ has no nontrivial automorphisms compatible with it, since $[-1]$ acts nontrivially on $A[n]$ for $n \geq 3$.

The Hilbert scheme $\mathrm{Hilb}^P_{X/S}$ is a fine moduli space for closed subschemes of $X$ with Hilbert polynomial $P$. This works because closed subschemes (as opposed to abstract varieties) have no nontrivial automorphisms: they are rigidified by their embedding.

The stack $\mathcal{B}un_n$ assigns to a scheme $T$ the groupoid of rank-$n$ vector bundles on $T$. This is a stack for the fpqc topology. For a vector bundle $\mathcal{E}$, the automorphism group is $\mathrm{GL}(\mathcal{E})$, which is never trivial, so $\mathcal{B}un_n$ is not representable.

The coarse moduli space is a single point $\mathrm{Spec}\, k$ (since all vector bundles of rank $n$ over $\mathrm{Spec}\, k$ are isomorphic), which is essentially useless. The stack, however, captures the rich geometry.

For genus 2 curves, every curve is hyperelliptic, so $\mathrm{Aut}(C) \supseteq \mathbb{Z}/2\mathbb{Z}$. The stack $\mathcal{M}_2$ is a smooth Deligne-Mumford stack of dimension 3. The coarse moduli space $M_2$ has quotient singularities at points corresponding to curves with extra automorphisms (e.g., the curve $y^2 = x^5 - x$ has $\mathrm{Aut}(C) \cong \mathbb{Z}/10\mathbb{Z}$, contributing a $\mathbb{Z}/5\mathbb{Z}$-singularity to $M_2$).

Consider $\mathbb{A}^{n+1} \setminus \lbrace 0 \rbrace$ with the $\mathbb{G}_m$-action given by weights $(w_0, \ldots, w_n)$: $\lambda \cdot (x_0, \ldots, x_n) = (\lambda^{w_0} x_0, \ldots, \lambda^{w_n} x_n).$ The quotient stack $[(\mathbb{A}^{n+1} \setminus \lbrace 0 \rbrace) / \mathbb{G}_m]$ is the stacky weighted projective space $\mathcal{P}(w_0, \ldots, w_n)$. When all $w_i = 1$, this is $\mathbb{P}^n$ (which is a scheme). For general weights, the coarse moduli space is the usual weighted projective space, which may have quotient singularities.

A family of conics in $\mathbb{P}^2$ parametrized by $T$ is a closed subscheme $\mathcal{C} \subset \mathbb{P}^2_T$ flat over $T$ with fibers being conics. The moduli space is $\mathbb{P}^5$ (the space of homogeneous quadratic forms in 3 variables up to scalar). Over the point $[a:b:c:d:e:f] \in \mathbb{P}^5$, the fiber is the conic $\lbrace ax^2 + bxy + cy^2 + dxz + eyz + fz^2 = 0 \rbrace$.

| Problem | Type | Reason | |---------|------|--------| | $\mathrm{Hilb}^P_{X/S}$ | Fine moduli scheme | No automorphisms (subschemes) | | $\mathrm{Gr}(k,n)$ | Fine moduli scheme | No automorphisms (quotients) | | $\mathcal{M}_g$ ($g \geq 2$) | DM stack | Finite automorphism groups | | $\mathcal{M}_{1,1}$ | DM stack | $\mathrm{Aut}(E) \supseteq \lbrace \pm 1 \rbrace$ | | $\mathcal{B}un_n(C)$ | Artin stack | $\mathrm{Aut}(\mathcal{E}) \supseteq \mathbb{G}_m$ | | $[*/G]$ | Artin stack | $\mathrm{Aut}(*) = G$ |