Skip to main content
MathOverflow

Return to Answer

deleted 1264 characters in body
Source Link
user19475
user19475

Let me sketch an idea why there are not enough Jacobians. At some points I have asked for references/proofs in []. Please help me to fill this gaps.

Let $X/k$ be a connected proper variety. We want to show that for $\dim{X} > 0$ not every Abelian scheme over $X$ is a quotient of a Jacobian of a (smooth proper) relative curve over $X$.

Motivation: Every smooth proper relative curve of genus $1$ over $X$ is isoconstant: After suitable base change with $K/k$ finite one can assume that $X$ has a $k$-rational point and level-$n$-structure with $n \geq 3$. But then $\mathcal{M}_{1,n}$ is a fine moduli space and affine, so every morphism $X \to \mathcal{M}_{1,n}$ is constant.

Now we generalise this to smooth proper relative curves of higher genusEdit:

The complement of $\mathcal{M}_{g,n}$ in the proper compactified moduli stack $\bar{\mathcal{M}}_{g,n}$ is a normal crossing divisor [reference?], and hence [is this true?] $\mathcal{M}_{g,n}$ is affine [what does this mean for a stack?]. Hence every smooth proper relative curve $\mathcal{C}/X$ is constant [is this true?].

Since $\mathrm{Pic}$ commutes with base change, every Jacobian of such a curve $\mathcal{C}/X$ is iso-constant. Hence also every quotient of a Jacobian of such a curve is iso-constant [is this true?]. But there are smooth proper subvarieties of $\mathcal{A}_{g,d,n}$ of dimension $> 0$ [is this true?] Didn't work out as hoped.

Let me sketch an idea why there are not enough Jacobians. At some points I have asked for references/proofs in []. Please help me to fill this gaps.

Let $X/k$ be a connected proper variety. We want to show that for $\dim{X} > 0$ not every Abelian scheme over $X$ is a quotient of a Jacobian of a (smooth proper) relative curve over $X$.

Motivation: Every smooth proper relative curve of genus $1$ over $X$ is isoconstant: After suitable base change with $K/k$ finite one can assume that $X$ has a $k$-rational point and level-$n$-structure with $n \geq 3$. But then $\mathcal{M}_{1,n}$ is a fine moduli space and affine, so every morphism $X \to \mathcal{M}_{1,n}$ is constant.

Now we generalise this to smooth proper relative curves of higher genus:

The complement of $\mathcal{M}_{g,n}$ in the proper compactified moduli stack $\bar{\mathcal{M}}_{g,n}$ is a normal crossing divisor [reference?], and hence [is this true?] $\mathcal{M}_{g,n}$ is affine [what does this mean for a stack?]. Hence every smooth proper relative curve $\mathcal{C}/X$ is constant [is this true?].

Since $\mathrm{Pic}$ commutes with base change, every Jacobian of such a curve $\mathcal{C}/X$ is iso-constant. Hence also every quotient of a Jacobian of such a curve is iso-constant [is this true?]. But there are smooth proper subvarieties of $\mathcal{A}_{g,d,n}$ of dimension $> 0$ [is this true?].

Let me sketch an idea why there are not enough Jacobians. At some points I have asked for references/proofs in []. Edit: Didn't work out as hoped.

Source Link
user19475
user19475

Let me sketch an idea why there are not enough Jacobians. At some points I have asked for references/proofs in []. Please help me to fill this gaps.

Let $X/k$ be a connected proper variety. We want to show that for $\dim{X} > 0$ not every Abelian scheme over $X$ is a quotient of a Jacobian of a (smooth proper) relative curve over $X$.

Motivation: Every smooth proper relative curve of genus $1$ over $X$ is isoconstant: After suitable base change with $K/k$ finite one can assume that $X$ has a $k$-rational point and level-$n$-structure with $n \geq 3$. But then $\mathcal{M}_{1,n}$ is a fine moduli space and affine, so every morphism $X \to \mathcal{M}_{1,n}$ is constant.

Now we generalise this to smooth proper relative curves of higher genus:

The complement of $\mathcal{M}_{g,n}$ in the proper compactified moduli stack $\bar{\mathcal{M}}_{g,n}$ is a normal crossing divisor [reference?], and hence [is this true?] $\mathcal{M}_{g,n}$ is affine [what does this mean for a stack?]. Hence every smooth proper relative curve $\mathcal{C}/X$ is constant [is this true?].

Since $\mathrm{Pic}$ commutes with base change, every Jacobian of such a curve $\mathcal{C}/X$ is iso-constant. Hence also every quotient of a Jacobian of such a curve is iso-constant [is this true?]. But there are smooth proper subvarieties of $\mathcal{A}_{g,d,n}$ of dimension $> 0$ [is this true?].