The Torelli map $\tau\colon M\_g \to A\_g$ sends a curve C to its Jacobian (along with the canonical principal polarization associated to C); see [this](https://mathoverflow.net/questions/7505/are-jacobians-principally-polarized-over-non-algebraically-closed-fields/7513#7513) question for a description which works for families. 



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> Theorem (Torelli): If $\tau(C) \cong \tau(C')$, then $C \cong C'$.
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If one prefers to work with coarse spaces (instead of stacks) it is okay to just say that $\tau$ is injective.



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> Question: Is $\tau$ an immersion?
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(One remark: $\tau$ isn't a closed immersion -- the closure of its image consists of products of Jacobians!)



Is the Torelli map an immersion?
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.


