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Respectfully, I disagree with Tony's answer. The infinitesimal Torelli problem fails for $g>2$ at the points of $M_g$ corresponding to the hyperelliptic curves. And in general the situation is trickier than one would expect.

The tangent space to the deformation space of a curve $C$ is $H^1(T_C)$, and the tangent space to the deformation space of its Jacobian is $Sym^2(H^1(\mathcal O_C))$. The infinitesimal Torelli map is an immersion iff the map of these tangent spaces

$$ H^1(T_C) \to Sym^2( H^1(\mathcal O_C) )$$

is an injection. Dually, the following map should be a surjection:

$$ Sym^2 ( H^0(K_C) ) \to H^0( 2K_C ), $$

where $K_C$ denotes the canonical class of the curve $C$. This is a surjection iff $g=1,2$ or $g=3$ and $C$ is not hyperelliptic; by a result of Max Noether.

Therefore, for $g\ge 3$ the Torelli map OF STACKS $\tau:M_g\to A_g$ is not an immersion. It is an immersion outside of the hyperelliptic locus $H_g$. Also, the restriction $\tau_{H_g}:H_g\to A_g$ is an immersion.

On the other hand, the Torelli map between the coarse moduli spaces IS an immersion in char 0. This is a result of Oort and Steenbrink "The local Torelli problem for algebraic curves" (1979).

F. Catanese gave a nice overview of the various flavors of Torelli maps (infinitesimal, local, global, generic) in "Infinitesimal Torelli problems and counterexamples to Torelli problems" (chapter 8 in "Topics in transcendental algebraic geometry" edited by Griffiths).

P.S. "Stacks" can be replaced everywhere by the "moduli spaces with level structure of level $l\ge3$ (which are fine moduli spaces).

P.P.S. The space of the first-order deformations of an abelian variety $A$ is $H^1(T_A)$. Since $T_A$ is a trivial vector bundle of rank $g$, and the tangent space at the origin is $H^0(\Omega^1_A)$, this space equals $H^1(\mathcal O_A)\otimes H^0(\Omega^1_A)$ and has dimension $g^2$.

A polarization $\lambda:A\to A^t$ is a homomorphism from $A$ to the dual abelian variety $A^t$. It gives a homomorphism on the tangent spaces to the origin, and for $A^t$ the tangent space is $H^1(\mathcal O_A)$. The subspace of first-order deformations preserving a polarization can be identified with $Sym^2(H^1(\mathcal O_A))$, and has dimension $g(g+1)/2$.

Respectfully, I disagree with Tony's answer. The infinitesimal Torelli problem fails for $g>2$ at the points of $M_g$ corresponding to the hyperelliptic curves. And in general the situation is trickier than one would expect.

The tangent space to the deformation space of a curve $C$ is $H^1(T_C)$, and the tangent space to the deformation space of its Jacobian is $Sym^2(H^1(\mathcal O_C))$. The infinitesimal Torelli map is an immersion iff the map of these tangent spaces

$$ H^1(T_C) \to Sym^2( H^1(\mathcal O_C) )$$

is an injection. Dually, the following map should be a surjection:

$$ Sym^2 ( H^0(K_C) ) \to H^0( 2K_C ), $$

where $K_C$ denotes the canonical class of the curve $C$. This is a surjection iff $g=1,2$ or $g=3$ and $C$ is not hyperelliptic; by a result of Max Noether.

Therefore, for $g\ge 3$ the Torelli map OF STACKS $\tau:M_g\to A_g$ is not an immersion. It is an immersion outside of the hyperelliptic locus $H_g$. Also, the restriction $\tau_{H_g}:H_g\to A_g$ is an immersion.

On the other hand, the Torelli map between the coarse moduli spaces IS an immersion in char 0. This is a result of Oort and Steenbrink "The local Torelli problem for algebraic curves" (1979).

F. Catanese gave a nice overview of the various flavors of Torelli maps (infinitesimal, local, global, generic) in "Infinitesimal Torelli problems and counterexamples to Torelli problems" (chapter 8 in "Topics in transcendental algebraic geometry" edited by Griffiths).

P.S. "Stacks" can be replaced everywhere by the "moduli spaces with level structure of level $l\ge3$ (which are fine moduli spaces).

P.P.S. The space of the first-order deformations of an abelian variety $A$ is $H^1(T_A)$. Since $T_A$ is a trivial vector bundle of rank $g$, and the cotangent space at the origin is $H^0(\Omega^1_A)$, this space equals $H^1(\mathcal O_A) \otimes H^0(\Omega^1_A)^{\vee}$ and has dimension $g^2$.

A polarization is a homomorphism $\lambda:A\to A^t$ from $A$ to the dual abelian variety $A^t$. It induces an isomorphism (in char 0, or for a principal polarization) from the tangent space at the origin $T_{A,0}=H^0(\Omega_A^1)^{\vee}$ to the tangent space at the origin $T_{A^t,0}=H^1(\mathcal O_A)$. This gives an isomorphism $$ H^1(\mathcal O_A) \otimes H^0(\Omega^1_A)^{\vee} \to H^1(\mathcal O_A) \otimes H^1(\mathcal O_A). $$

The subspace of first-order deformations which preserve the polarization $\lambda$ can be identified with the tensors mapping to zero in $\wedge^2 H^1(\mathcal O_A)$, and so is isomorphic to $Sym^2 H^1(\mathcal O_A)$, outside of characteristic 2.

Respectfully, I disagree with Tony's answer. The infinitesimal Torelli problem fails for $g>2$ at the points of $M_g$ corresponding to the hyperelliptic curves. And in general the situation is trickier than one would expect.

The tangent space to the deformation space of a curve $C$ is $H^1(T_C)$, and the tangent space to the deformation space of its Jacobian is $Sym^2(H^1(\mathcal O_C))$. The infinitesimal Torelli map is an immersion iff the map of these tangent spaces

$$ H^1(T_C) \to Sym^2( H^1(\mathcal O_C) )$$

is an injection. Dually, the following map should be a surjection:

$$ Sym^2 ( H^0(K_C) ) \to H^0( 2K_C ), $$

where $K_C$ denotes the canonical class of the curve $C$. This is a surjection iff $g=1,2$ or $g=3$ and $C$ is not hyperelliptic; by a result of Max Noether.

Therefore, for $g\ge 3$ the Torelli map OF STACKS $\tau:M_g\to A_g$ is not an immersion. It is an immersion outside of the hyperelliptic locus $H_g$. Also, the restriction $\tau_{H_g}:H_g\to A_g$ is an immersion.

On the other hand, the Torelli map between the coarse moduli spaces IS an immersion in char 0. This is a result of Oort and Steenbrink "The local Torelli problem for algebraic curves" (1979).

F. Catanese gave a nice overview of the various flavors of Torelli maps (infinitesimal, local, global, generic) in "Infinitesimal Torelli problems and counterexamples to Torelli problems" (chapter 8 in "Topics in transcendental algebraic geometry" edited by Griffiths).

Respectfully, I disagree with Tony's answer. The infinitesimal Torelli problem fails for $g>2$ at the points of $M_g$ corresponding to the hyperelliptic curves. And in general the situation is trickier than one would expect.

The tangent space to the deformation space of a curve $C$ is $H^1(T_C)$, and the tangent space to the deformation space of its Jacobian is $Sym^2(H^1(\mathcal O_C))$. The infinitesimal Torelli map is an immersion iff the map of these tangent spaces

$$ H^1(T_C) \to Sym^2( H^1(\mathcal O_C) )$$

is an injection. Dually, the following map should be a surjection:

$$ Sym^2 ( H^0(K_C) ) \to H^0( 2K_C ), $$

where $K_C$ denotes the canonical class of the curve $C$. This is a surjection iff $g=1,2$ or $g=3$ and $C$ is not hyperelliptic; by a result of Max Noether.

Therefore, for $g\ge 3$ the Torelli map OF STACKS $\tau:M_g\to A_g$ is not an immersion. It is an immersion outside of the hyperelliptic locus $H_g$. Also, the restriction $\tau_{H_g}:H_g\to A_g$ is an immersion.

On the other hand, the Torelli map between the coarse moduli spaces IS an immersion in char 0. This is a result of Oort and Steenbrink "The local Torelli problem for algebraic curves" (1979).

F. Catanese gave a nice overview of the various flavors of Torelli maps (infinitesimal, local, global, generic) in "Infinitesimal Torelli problems and counterexamples to Torelli problems" (chapter 8 in "Topics in transcendental algebraic geometry" edited by Griffiths).

P.S. "Stacks" can be replaced everywhere by the "moduli spaces with level structure of level $l\ge3$ (which are fine moduli spaces).

P.P.S. The space of the first-order deformations of an abelian variety $A$ is $H^1(T_A)$. Since $T_A$ is a trivial vector bundle of rank $g$, and the tangent space at the origin is $H^0(\Omega^1_A)$, this space equals $H^1(\mathcal O_A)\otimes H^0(\Omega^1_A)$ and has dimension $g^2$.

A polarization $\lambda:A\to A^t$ is a homomorphism from $A$ to the dual abelian variety $A^t$. It gives a homomorphism on the tangent spaces to the origin, and for $A^t$ the tangent space is $H^1(\mathcal O_A)$. The subspace of first-order deformations preserving a polarization can be identified with $Sym^2(H^1(\mathcal O_A))$, and has dimension $g(g+1)/2$.

Respectfully, I disagree with Tony's answer. The infinitesimal Torelli problem fails for $g>2$ at the points of $M_g$ corresponding to the hyperelliptic curves. And in general the situation is rather tricky.

Griffiths' result shows that on the tangent spaces to deformation spaces the Torelli map is dual to

$$ Sym^2 (H^0(C,\Omega^1_C)) \to H^0( C, (\Omega^1_C)^{\otimes 2} ) $$

This is not surjective precisely when $g\ge 3$ and $C$ is hyperelliptic. Indeed, failure of infinitesimal Torelli is a classical result of M. Noether.

Therefore, the Torelli map OF STACKS $\tau:M_g\to A_g$ is not an immersion. It is an immersion outside of the hyperelliptic locus $H_g$. Also, I think the restriction $\tau_{H_g}:H_g\to A_g$ is an immersion.

On the other hand, the Torelli map between coarse moduli spaces IS an immersion in char 0. This is a result of Oort and Steenbrink "The local Torelli problem for algebraic curves" (1979).

F. Catanese gave a nice overview of the various flavors of Torelli maps (infinitesimal, local, global, generic) in "Infinitesimal Torelli problems and counterexamples to Torelli problems" (chapter 8 in "Topics in transcendental algebraic geometry" edited by Griffiths).

Respectfully, I disagree with Tony's answer. The infinitesimal Torelli problem fails for $g>2$ at the points of $M_g$ corresponding to the hyperelliptic curves. And in general the situation is trickier than one would expect.

The tangent space to the deformation space of a curve $C$ is $H^1(T_C)$, and the tangent space to the deformation space of its Jacobian is $Sym^2(H^1(\mathcal O_C))$. The infinitesimal Torelli map is an immersion iff the map of these tangent spaces

$$ H^1(T_C) \to Sym^2( H^1(\mathcal O_C) )$$

is an injection. Dually, the following map should be a surjection:

$$ Sym^2 ( H^0(K_C) ) \to H^0( 2K_C ), $$

where $K_C$ denotes the canonical class of the curve $C$. This is a surjection iff $g=1,2$ or $g=3$ and $C$ is not hyperelliptic; by a result of Max Noether.

Therefore, for $g\ge 3$ the Torelli map OF STACKS $\tau:M_g\to A_g$ is not an immersion. It is an immersion outside of the hyperelliptic locus $H_g$. Also, the restriction $\tau_{H_g}:H_g\to A_g$ is an immersion.

On the other hand, the Torelli map between the coarse moduli spaces IS an immersion in char 0. This is a result of Oort and Steenbrink "The local Torelli problem for algebraic curves" (1979).

F. Catanese gave a nice overview of the various flavors of Torelli maps (infinitesimal, local, global, generic) in "Infinitesimal Torelli problems and counterexamples to Torelli problems" (chapter 8 in "Topics in transcendental algebraic geometry" edited by Griffiths).