04.10.34 · algebraic-geometry / moduli

Torelli morphism and Torelli theorem

shipped3 tiersLean: none

Anchor (Master): Harris-Morrison Ch. 6 (Springer GTM 187, 1998); Arbarello-Cornalba-Griffiths-Harris *Geometry of Algebraic Curves* Vol. II (Springer Grundlehren 268, 2011); Andreotti 1958 *Amer. J. Math.* 80 (proof via theta singularities); Torelli 1913 *Rend. Circ. Mat. Palermo* 36 (original statement); Mumford *Tata Lectures on Theta* I, II (Birkhäuser 1983, 1984); Beauville 1987 *Astérisque* 145-146 (Schottky); Welters 1986 *Acta Math.* 157 (trisecant); Clemens-Griffiths 1972 *Ann. of Math.* 95 (cubic threefolds); Piatetski-Shapiro-Shafarevich 1971 *Izv. Akad. Nauk SSSR* 35 (K3 Torelli)

Intuition Beginner

A smooth projective curve $C$ of genus $g$ carries a finite-dimensional space of holomorphic $1$ -forms, of dimension $g$ . Integrating each form around each loop in the curve produces a $g$ -dimensional lattice of periods. The quotient of $g$ -dimensional complex space by this lattice is the Jacobian $J (C)$ , a compact complex torus that packages all of the curve's period data into one geometric object. The Jacobian also carries a special divisor called the theta divisor $Θ$ , which records how the curve sits inside its own period torus.

The Torelli theorem of 1913 says something remarkable. The pair (Jacobian, theta divisor) determines the curve. Two smooth curves with isomorphic Jacobians and matching theta divisors must be isomorphic as curves. So the lattice of periods, together with the extra theta-divisor data, remembers everything about the original curve. Knowing the periods is the same as knowing the curve.

Why bother? Because the Jacobian lives in a much larger world: the moduli space of principally polarised abelian varieties of dimension $g$ , written $A_{g}$ . The Torelli theorem is the statement that the assignment "curve to Jacobian" is an injection from the moduli of curves to the moduli of polarised abelian varieties. This injection lets one study curves through linear-algebra data on their Jacobians, and it lets one ask the converse question: which polarised abelian varieties arise as Jacobians of curves? The latter is the Schottky problem, one of the central open questions of nineteenth-century mathematics, fully solved in genus $4$ and still active today.

Visual Beginner

A schematic of the moduli space $M_{g}$ of smooth curves on the left and the moduli space $A_{g}$ of principally polarised abelian varieties on the right, with an arrow labelled $τ$ going from left to right. A typical point on the left, a curve $C$ , is sent by $τ$ to a point on the right, the polarised pair $(J (C), Θ)$ . A second panel shows the theta divisor $Θ \subset J (C)$ as a hypersurface with a marked singular locus, and notes that the singular locus reconstructs the canonical curve sitting inside the Jacobian — the key to Andreotti's proof.

The picture captures the essential geometry: a curve is sent to a polarised abelian variety, and the polarisation records enough internal structure of the Jacobian to recover the curve. In low genus the map is almost surjective, and in high genus the image is a thin subvariety whose defining equations are the Schottky problem.

Worked example Beginner

Compute the Torelli image for the lowest genera and check the dimension count against the Torelli theorem.

Step 1. Genus $g = 1$ . A smooth genus- $1$ curve is an elliptic curve $E$ . The Jacobian is $J (E) = E$ itself, and the theta divisor is a single point. The moduli space of curves $M_{1}$ has dimension $1$ , parametrised by the $j$ -invariant. The moduli space of principally polarised abelian varieties $A_{1}$ also has dimension $1$ , parametrised by the same $j$ -invariant. The Torelli morphism $τ : M_{1} \to A_{1}$ is the identity on the level of coarse moduli spaces, so the Torelli theorem holds in genus $1$ by the construction itself.

Step 2. Genus $g = 2$ . Every smooth genus- $2$ curve $C$ is hyperelliptic. The dimension of $M_{2}$ is $3 \cdot 2 - 3 = 3$ , and the dimension of $A_{2}$ is $2 \cdot 3/2 = 3$ . The two dimensions match, and the Torelli morphism is a birational map onto its image. A classical theorem says that every indecomposable principally polarised abelian surface arises as the Jacobian of a smooth genus- $2$ curve, so the Torelli image is open in $A_{2}$ .

Step 3. Genus $g = 3$ . The dimension of $M_{3}$ is $3 \cdot 3 - 3 = 6$ , and the dimension of $A_{3}$ is $3 \cdot 4/2 = 6$ . Again the dimensions match, and the Torelli morphism is generically an open immersion. The complement of the Torelli image in $A_{3}$ has positive codimension and is filled by products of lower-dimensional Jacobians.

Step 4. Genus $g = 4$ . The dimension of $M_{4}$ is $3 \cdot 4 - 3 = 9$ , and the dimension of $A_{4}$ is $4 \cdot 5/2 = 10$ . Now the source has strictly smaller dimension than the target, and the Torelli image is a hypersurface in $A_{4}$ . The defining equation of this hypersurface was found by Schottky in 1888 — a single modular form of weight $8$ that vanishes precisely on the closure of the Jacobian locus.

What this tells us: the Torelli morphism is an injection of stacks for every $g$ , but it is only generically open in genera $2$ and $3$ . From genus $4$ onward the image is a thin subvariety whose codimension grows. The numerical mismatch $dim A_{g} - dim M_{g} = g (g + 1) /2 - (3 g - 3) = (g - 2) (g - 3) /2$ measures the codimension of the Jacobian locus, and computing equations for that locus is the modern Schottky problem.

Check your understanding Beginner

Exercise (easy, multiple choice).

The Torelli theorem states that the assignment $[C] \mapsto [J (C), Θ]$ from smooth projective curves to principally polarised abelian varieties is:

A. Surjective for every $g$
B. Injective on isomorphism classes of curves, with the polarisation determining the curve
C. An isomorphism of moduli spaces for every $g$
D. Constant on hyperelliptic curves

Hint

The Torelli theorem says that the pair (Jacobian, theta divisor) recovers the curve up to isomorphism. It does not say the assignment is onto, and it does not say the moduli spaces match in dimension.

Answer

B. The Torelli theorem of 1913 says that if $C_{1}$ and $C_{2}$ are smooth projective curves with $(J (C_{1}), Θ_{1}) ≅ (J (C_{2}), Θ_{2})$ as principally polarised abelian varieties, then $C_{1} ≅ C_{2}$ as curves. This is an injectivity statement on isomorphism classes, with the polarisation playing the load-bearing role. The Torelli morphism is not surjective onto $A_{g}$ for $g \geq 4$ — that is the Schottky problem.

Formal definition Intermediate+

Let $k = C$ (the constructions extend to other algebraically closed fields with the appropriate analytic-versus-algebraic translation). Let $C$ be a smooth projective curve of genus $g \geq 1$ .

Definition (Jacobian variety). The Jacobian of $C$ is the principally polarised abelian variety $$ J(C) = H^0(C, \omega_C)^\vee / H_1(C, \mathbb{Z}), $$ where $ω_{C} = Ω_{C}^{1}$ is the canonical sheaf, $H^{0} (C, ω_{C})$ is the $g$ -dimensional space of holomorphic $1$ -forms (see 04.09.01), and $H_{1} (C, Z) ↪ H^{0} (C, ω_{C})^{\lor}$ is the lattice obtained by sending a $1$ -cycle $γ$ to the linear functional $ω \mapsto \int_{γ} ω$ . The quotient $J (C)$ is a compact complex torus of dimension $g$ , and Riemann's bilinear relations together with the cup-product polarisation produce on $J (C)$ a positive line bundle $L_{Θ}$ with $h^{0} (J (C), L_{Θ}) = 1$ . The pair $(J (C), [L_{Θ}])$ is a principally polarised abelian variety (PPAV) of dimension $g$ .

Definition (theta divisor). The theta divisor $Θ \subset J (C)$ is the unique-up-to-translation effective divisor whose line bundle is $L_{Θ}$ . Equivalently, fixing a base point $p_{0} \in C$ and writing $AJ : C^{g - 1} \to J (C)$ for the Abel-Jacobi map sending $(p_{1}, \dots, p_{g - 1})$ to the class of $O_{C} (p_{1} + \dots + p_{g - 1} - (g - 1) p_{0})$ in $Pic^{0} (C) ≅ J (C)$ (see 04.05.02), the theta divisor is the image $$ \Theta = \mathrm{AJ}\big(C^{g - 1}\big) \subset J(C), $$ an irreducible hypersurface of dimension $g - 1$ .

Definition (Brill-Noether locus $W_{d}^{r}$ ). The Brill-Noether locus $W_{d}^{r} (C) \subset Pic^{d} (C)$ is the closed subscheme parametrising line bundles $L$ of degree $d$ on $C$ with $h^{0} (L) \geq r + 1$ . In particular $W_{g - 1}^{0} (C) = Θ$ (under the identification $Pic^{g - 1} (C) ≅ J (C)$ via a chosen theta-characteristic), and $W_{g - 1}^{1} (C)$ is the locus of degree- $(g - 1)$ line bundles with at least $2$ independent global sections.

Definition (moduli space of PPAVs). The moduli space $A_{g}$ of principally polarised abelian varieties of dimension $g$ is the quotient $H_{g} / Sp (2 g, Z)$ of the Siegel upper half space $$ \mathfrak{H}g = {\tau \in \mathrm{Mat}{g \times g}(\mathbb{C}) : \tau = \tau^T, \mathrm{Im},\tau > 0} $$ by the integral symplectic group acting by fractional linear transformations $τ \mapsto (A τ + B) (C τ + D)^{- 1}$ . As a complex orbifold (and a Deligne-Mumford stack), $A_{g}$ has dimension $g (g + 1) /2$ .

Definition (Torelli morphism). The Torelli morphism is the morphism of moduli stacks $$ \tau : \mathcal{M}_g \to \mathcal{A}_g, \qquad [C] \mapsto [J(C), \Theta], $$ sending the isomorphism class of a smooth curve $C$ to the isomorphism class of its principally polarised Jacobian. The image is the Jacobian locus $J_{g} = τ (M_{g})$ .

Counterexamples to common slips

The Jacobian alone, without the polarisation, does not determine the curve. Two non-isomorphic curves can have isomorphic Jacobians as complex tori; what fails is that the isomorphism does not respect the theta divisors. The polarisation is load-bearing.
The Torelli morphism is an injection on geometric points but not in general a closed immersion of stacks. The hyperelliptic locus and the locus of curves with extra automorphisms create stacky subtleties; the geometric-point statement is clean, the stack-theoretic statement requires care.
The Riemann singularity theorem $mult_{x} Θ = h^{0} (L_{x})$ at a point $x \in Θ$ corresponding to a line bundle $L_{x}$ of degree $g - 1$ is the bridge from the theta divisor to Brill-Noether geometry; treating $Θ$ as a smooth hypersurface misses the entire reconstruction story.
The Schottky problem is not asking which complex tori arise as Jacobians; complex tori without polarisation are far easier to classify. The problem is which principally polarised abelian varieties arise, and this is where the rich modular-forms equations enter.

Key theorem with proof Intermediate+

Theorem (Torelli, 1913; Andreotti, 1958). Let $C_{1}$ and $C_{2}$ be smooth projective curves of genus $g \geq 2$ over an algebraically closed field of characteristic zero. If there exists an isomorphism $(J (C_{1}), Θ_{1}) ≅ (J (C_{2}), Θ_{2})$ of principally polarised abelian varieties, then $C_{1} ≅ C_{2}$ as curves. Equivalently, the Torelli morphism $τ : M_{g} \to A_{g}$ is injective on geometric points.

Proof (Andreotti's reconstruction; non-hyperelliptic case). The argument reconstructs the canonical model of $C$ inside $J (C)$ from the singular locus of $Θ$ , then identifies it as the canonical curve via projective duality. The argument has four steps: identify the singular locus of $Θ$ with the Brill-Noether locus $W_{g - 1}^{1} (C)$ , exhibit the Gauss map on $Θ$ , project to recover the canonical curve, and conclude isomorphism of curves from isomorphism of canonical models.

Step 1: Riemann singularity theorem. Translate $Θ$ so it equals the Abel-Jacobi image of $C^{g - 1}$ in $J (C) ≅ Pic^{g - 1} (C)$ . The Riemann singularity theorem says that at a point $x \in Θ$ corresponding to a line bundle $L_{x}$ of degree $g - 1$ , $$ \mathrm{mult}x \Theta = h^0(C, L_x), $$ so $x \in Θ$ iff $h^{0} (L_{x}) \geq 1$ and $x \in Sing (Θ)$ iff $h^{0} (L_{x}) \geq 2$ . Therefore the singular locus of the theta divisor reads as the Brill-Noether locus, $$ \mathrm{Sing}(\Theta) = W^1{g - 1}(C), $$ the locus of degree- $(g - 1)$ line bundles with at least two independent global sections. The Riemann singularity theorem is itself a calculation in local cohomology, comparing the multiplicity of $Θ$ along the Abel-Jacobi image of $C^{g - 1}$ to the rank of the differential of the Abel-Jacobi map, which is the rank of the evaluation map $H^{0} (L_{x}) \otimes H^{0} (ω_{C} \otimes L_{x}^{- 1}) \to H^{0} (ω_{C})$ at the point $L_{x}$ .

Step 2: Gauss map on $Θ$ . Each smooth point $y$ of $Θ$ has a well-defined tangent space $T_{y} Θ \subset T_{y} J (C) ≅ H^{0} (C, ω_{C})^{\lor}$ , a hyperplane. The Gauss map of $Θ$ is the morphism $$ \mathcal{G} : \Theta_{\mathrm{sm}} \to \mathbb{P}\big(H^0(C, \omega_C)\big) = \mathbb{P}^{g - 1} $$ sending a smooth point $y$ to the projectivisation of the conormal direction $T_{y} Θ^{⊥} \subset H^{0} (C, ω_{C})$ . The Gauss map is rational on all of $Θ$ and regular on $Θ_{sm}$ .

Step 3: image of $G$ . A direct calculation in coordinates (Andreotti 1958 §3) shows that for a non-hyperelliptic curve $C$ , the image $G (Θ_{sm})$ is contained in the canonical curve $$ \phi_K(C) \subset \mathbb{P}\big(H^0(C, \omega_C)\big) = \mathbb{P}^{g - 1}, $$ namely the image of $C$ under the canonical embedding $ϕ_{K} : C \to P^{g - 1}$ , and in fact $G$ degree- $(g - 1 2 g - 2)$ -covers $ϕ_{K} (C)$ . Equivalently, the branch locus of $G$ , the dual variety of $ϕ_{K} (C)$ , recovers $ϕ_{K} (C)$ via projective duality of curves in $P^{g - 1}$ . So from the polarised pair $(J (C), Θ)$ one constructs $ϕ_{K} (C) \subset P^{g - 1}$ purely from the Gauss-map branch locus.

Step 4: canonical model recovers the curve. A non-hyperelliptic smooth projective curve $C$ of genus $g \geq 3$ is determined up to isomorphism by its canonical model $ϕ_{K} (C) \subset P^{g - 1}$ , since $ϕ_{K} : C \to ϕ_{K} (C)$ is an embedding. Given an isomorphism $(J (C_{1}), Θ_{1}) ≅ (J (C_{2}), Θ_{2})$ , the construction of the previous step produces an isomorphism $ϕ_{K} (C_{1}) ≅ ϕ_{K} (C_{2})$ inside $P^{g - 1}$ , and hence $C_{1} ≅ C_{2}$ as curves.

Proof, hyperelliptic case. A smooth genus- $g \geq 2$ curve $C$ is hyperelliptic if it admits a degree- $2$ map to $P^{1}$ , equivalently if $ϕ_{K} : C \to P^{g - 1}$ is a degree- $2$ map onto a rational normal curve and not an embedding. In this case the Gauss-map calculation of Step 3 gives a rational normal curve in $P^{g - 1}$ , which does not recover $C$ directly. A separate argument (Andreotti 1958 §5; reformulated in Arbarello-Cornalba-Griffiths-Harris XI §6) recovers $C$ from the $2 g + 2$ Weierstrass points, identified as the fixed points of the hyperelliptic involution on $J (C)$ acting by $- 1$ . The hyperelliptic involution on the Jacobian is induced by the unique nontrivial involution of the polarised pair $(J (C), Θ)$ in the hyperelliptic case, and its fixed locus on $C$ is the $2 g + 2$ Weierstrass points. The cross-ratio data of these points on $P^{1}$ determines $C$ up to isomorphism, completing the proof. $□$

Bridge. The Torelli theorem builds toward the Schottky problem and the geometry of the moduli space $A_{g}$ , and the central insight is that the singular locus of the theta divisor reconstructs the canonical curve, identifying the curve with a projective-dual construction inside its own Jacobian. This bridge appears again in 04.10.01 (moduli of curves), where the Torelli morphism realises $M_{g}$ as a stack inside $A_{g}$ and the foundational reason for the dimension count $dim M_{g} = 3 g - 3$ versus $dim A_{g} = g (g + 1) /2$ is recorded in the codimension of the Jacobian locus. The bridge is that the Riemann singularity theorem $Sing (Θ) = W_{g - 1}^{1}$ identifies the singular geometry of the theta divisor with the Brill-Noether geometry of special line bundles on the curve, and this identifies $J (C)$ with a curve invariant whose singular polarised data is the Brill-Noether stratification of $Pic^{g - 1} (C)$ .

Putting these together, the Torelli morphism is the link between two parallel moduli theories: the moduli of curves controlled by Riemann-Roch and deformation theory on the source side, and the moduli of polarised abelian varieties controlled by theta functions and the Siegel modular group on the target side. The Schottky problem is exactly the problem of cutting out the Jacobian locus by explicit modular equations, and Welters's trisecant identity (1986), proved by Krichever in 2010, generalises the Schottky relation: the Kummer variety of a PPAV $(A, Θ)$ admits a trisecant line iff $A$ is a Jacobian. This pattern recurs in the Clemens-Griffiths cubic-threefold proof of generic Torelli for cubic threefolds, where the intermediate Jacobian plays the role of $J (C)$ and the analogous singular-locus reconstruction succeeds.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Let $C$ be a smooth genus- $2$ curve. Show that the theta divisor $Θ \subset J (C)$ is a smooth curve isomorphic to $C$ .

Hint

For $g = 2$ , the theta divisor is the Abel-Jacobi image of $C^{g - 1} = C^{1} = C$ . The Abel-Jacobi map $C \to J (C)$ is an embedding when $g \geq 1$ and a base point is fixed.

Answer

For $g = 2$ , the Abel-Jacobi map $AJ : C \to J (C)$ , $p \mapsto [O_{C} (p - p_{0})]$ , is an embedding (it is injective by Riemann-Roch on a smooth curve of genus $\geq 1$ , and unramified since $C$ is smooth). The theta divisor is the image $Θ = AJ (C^{g - 1}) = AJ (C)$ . So $Θ ≅ C$ as abstract curves, and $Θ$ is smooth since $C$ is smooth. The Riemann singularity theorem $mult_{x} Θ = h^{0} (L_{x})$ confirms that $Θ$ is everywhere smooth: at a point $L_{x}$ of degree $1$ , the multiplicity is $h^{0} (L_{x})$ which equals $1$ generically (and equals $1$ throughout the image since $g - 1 = 1$ does not allow Brill-Noether jumps for line bundles of degree $1$ on a genus- $2$ curve). Rubric: full credit for $Θ ≅ C$ with the embedding argument and the Riemann-singularity confirmation of smoothness.

Exercise 4 (medium, symbolic).

Compute the dimension of $W_{g - 1}^{1} (C)$ for a generic smooth projective curve $C$ of genus $g \geq 3$ , and identify when $Sing (Θ)$ is non-empty.

Hint

The expected dimension of $W_{d}^{r} (C)$ is given by the Brill-Noether number $ρ (g, r, d) = g - (r + 1) (g - d + r)$ . For $W_{g - 1}^{1}$ , this is $ρ = g - 2 (g - (g - 1) + 1) = g - 4$ .

Answer

$dim W_{g - 1}^{1} (C) = g - 4$ generically, non-empty iff $g \geq 4$ . The Brill-Noether number is $ρ (g, 1, g - 1) = g - 2 \cdot 2 = g - 4$ (substituting $r = 1$ , $d = g - 1$ ). The Brill-Noether theorem of Griffiths-Harris-Kempf-Kleiman-Laksov says that for a generic curve $C$ of genus $g$ , the locus $W_{d}^{r}$ is non-empty of dimension $ρ$ when $ρ \geq 0$ , and is empty when $ρ < 0$ . So $W_{g - 1}^{1} (C)$ is empty for $g \leq 3$ and has dimension $g - 4$ for $g \geq 4$ generically. By the Riemann singularity theorem $Sing (Θ) = W_{g - 1}^{1}$ , this says $Θ$ is smooth on a generic genus- $2$ or genus- $3$ curve and has a $(g - 4)$ -dimensional singular locus on a generic curve of genus $\geq 4$ . Rubric: full credit for $g - 4$ with the Brill-Noether-number derivation.

Exercise 5 (medium, short-answer).

State the Schottky relation in genus $4$ , and explain its role in the Schottky problem.

Hint

Schottky 1888 wrote down a single polynomial relation among the theta-nullwerte at second-order characteristics on the Siegel space $H_{4}$ . The relation vanishes on Jacobians.

Answer

Schottky 1888 ^{[Schottky 1888]} wrote down a polynomial $J (τ)$ in the second-order theta constants $θ_{[a, b]} (0; τ)$ at half-integer characteristics $[a, b] \in (\frac{1}{2} Z / Z)^{8}$ on the Siegel upper half space $H_{4}$ such that $J (τ) = 0$ whenever $τ$ is the period matrix of a smooth genus- $4$ curve. Concretely $J$ is built from the sixteen second-order theta values via the modular relation $r_{1} + r_{2} + r_{3} = 0$ among certain combinations of products $\prod θ_{[a, b]}$ that classify the period matrices of Jacobians. Schottky and later Schottky-Jung conjectured that $J = 0$ cuts out the closure of the Jacobian locus $J_{4}$ inside $A_{4}$ , and Igusa 1981 ^{[Igusa 1981]} proved this conjecture: $\overline{J_{4}}$ is exactly the zero locus of $J$ , a hypersurface in $A_{4}$ defined by one modular form of weight $8$ . This is the original solution to the Schottky problem in genus $4$ . For higher genus, no analogous one-equation answer is known, but multiple alternative characterisations (Andreotti-Mayer, Novikov-Krichever, Buser-Sarnak) describe the Jacobian locus. Rubric: full credit for the modular-relation framing with the Igusa 1981 attribution.

Exercise 6 (medium, short-answer).

Outline the Andreotti-Weil 1952 infinitesimal Torelli theorem and explain how it differs from the global Torelli theorem.

Hint

Infinitesimal Torelli is the statement that the differential of $τ : M_{g} \to A_{g}$ is injective at every smooth-curve point. Global Torelli is the statement that $τ$ itself is injective on geometric points.

Answer

Infinitesimal Torelli (Andreotti-Weil 1952) ^{[Andreotti-Weil 1952]}. The differential of the Torelli morphism at a smooth curve $[C]$ is the linear map $$ d\tau_{[C]} : T_{[C]} \mathcal{M}g = H^1(C, T_C) \to T{[J(C), \Theta]} \mathcal{A}_g = \mathrm{Sym}^2 H^1(C, \mathcal{O}C), $$ identified via Serre duality $H^{1} (C, T_{C}) ≅ H^{0} (C, ω_{C}^{\otimes 2})^{\lor}$ with the dual of the multiplication map $Sym^{2} H^{0} (C, ω_{C}) \to H^{0} (C, ω_{C}^{\otimes 2})$ . Infinitesimal Torelli says that $d\tau{[C]} $i s inj ec t i v e f or e v er y$ [C] \in \mathcal{M}_g $, e q u i v a l e n tl y t ha tt h e m u l t i pl i c a t i o nma p$ \mathrm{Sym}^2 H^0(C, \omega_C) \to H^0(C, \omega_C^{\otimes 2}) $i s * s u r j ec t i v e * . T h e l a tt er i se x a c tl y t h es t a t e m e n t o f M a x N oe t h e r^{'} s t h eor e m o n c an o ni c a l c u r v eso f$ g \geq 3 $(f ai l in g o n l y o nh y p er e l l i pt i cs, w h er e in f ini t es ima l T or e l l ih o l d s b u t r e q u i r es t h e h y p er e l l i pt i c in v o l u t i o nana l y s i s), an d i t i s t h e d u a l r e f or m u l a t i o n t ha t A n d r eo tt ian d W e i l p r o v e d in 1952. Gl o ba l T or e l l i (T or e l l i 1913, A n d r eo tt i 1958) i s t h es t r o n g er s t a t e m e n tt ha t$ \tau$ itself is injective on geometric points, not just on tangent vectors. Infinitesimal Torelli holds in many situations where global Torelli fails (e.g., higher-dimensional varieties of general type), and conversely global Torelli sometimes holds without infinitesimal Torelli at every point (e.g., on certain boundary loci). The two statements are independent in general but coincide in spirit for smooth families. Rubric: full credit for the tangent-space identification, the multiplication-map formulation, and the distinction from global Torelli.

Exercise 7 (hard, short-answer).

Prove that every indecomposable principally polarised abelian surface arises as the Jacobian of a smooth genus- $2$ curve.

Hint

Show that the theta divisor of an indecomposable PPAV of dimension $2$ is a smooth curve of genus $2$ , and reconstruct the surface as its Jacobian.

Answer

Let $(A, Θ)$ be an indecomposable principally polarised abelian surface (i.e., $A$ is not isomorphic to a product of two elliptic curves with the product polarisation). Then $Θ \subset A$ is an irreducible effective divisor with $Θ^{2} = 2$ , and $Θ \cdot Θ = 2$ is the self-intersection coming from the unimodular principal polarisation. The adjunction formula on the abelian surface $A$ (with $K_{A} = 0$ ) gives $2 g_{Θ} - 2 = K_{A} \cdot Θ + Θ^{2} = 0 + 2 = 2$ , hence $g_{Θ} = 2$ . So $Θ$ is a smooth (since $A$ is indecomposable; for decomposable PPAVs the theta divisor would be reducible) projective curve of genus $2$ . The Abel-Jacobi map $Θ ↪ A$ induces a morphism of PPAVs $J (Θ) \to A$ which is an isomorphism by the universal property of the Jacobian: the inclusion $Θ \subset A$ factors uniquely through the Albanese variety of $Θ$ , which is $J (Θ)$ , and the resulting map is an isomorphism since both sides have dimension $2$ and the polarisations match by construction of $Θ$ . Hence $(A, Θ) ≅ (J (Θ), Θ_{J (Θ)})$ is the Jacobian of the genus- $2$ curve $Θ$ . Rubric: full credit for the adjunction computation, the universal-property argument, and the indecomposability hypothesis.

Exercise 8 (hard, short-answer).

Outline the Clemens-Griffiths 1972 proof that a smooth cubic threefold $X \subset P^{4}$ is not rational, and explain the role of the intermediate Jacobian and Torelli.

Hint

The intermediate Jacobian $J (X) = H^{2, 1} (X)^{\lor} / H_{3} (X, Z)$ is a $5$ -dimensional principally polarised abelian variety. Clemens and Griffiths showed that it is not a Jacobian of a smooth curve, while every rational threefold has intermediate Jacobian that decomposes as a Jacobian.

Answer

Clemens-Griffiths 1972 ^{[Clemens-Griffiths 1972]}. The intermediate Jacobian of a smooth complex projective threefold $X$ is the principally polarised abelian variety $$ J(X) = H^{2, 1}(X)^\vee / H_3(X, \mathbb{Z}), $$ of dimension $h^{2, 1} (X) = dim H^{2, 1} (X)$ . For a smooth cubic threefold $X \subset P^{4}$ a standard Hodge-number computation gives $h^{2, 1} (X) = 5$ , so $J (X)$ is a $5$ -dimensional PPAV.

Generic Torelli for cubic threefolds (Clemens-Griffiths 1972): the assignment $[X] \mapsto [J (X), Θ_{X}]$ is generically injective on the moduli space of smooth cubic threefolds, with the polarisation determined by the cup-product pairing on $H^{3} (X, Z)$ . The argument follows Andreotti's pattern: a singular-locus reconstruction recovers the Fano surface of lines on $X$ from $Sing (Θ_{X})$ , and this reconstructs the cubic threefold.

Irrationality consequence: a rational threefold $X$ has $J (X)$ isomorphic, as PPAV, to a product of Jacobians of smooth curves (Clemens-Griffiths proved this from the blow-up formula for intermediate Jacobians and the fact that birational invariance + blow-up of curves contributes Jacobian factors). But the generic-Torelli reconstruction shows that $J (X)$ for $X$ a smooth cubic threefold is not a product of Jacobians of curves: its singular-theta locus has the wrong structure, namely it is the Fano surface of lines, a smooth surface of general type, rather than a union of points or products of lower-dimensional Jacobian theta singularities. Hence $X$ is not rational. This was the first non-rationality result for a smooth Fano threefold of Picard rank $1$ , and it founded the modern theory of birational rigidity of Fano varieties. Rubric: full credit for the intermediate-Jacobian construction, the singular-locus reconstruction, and the non-rationality conclusion.

Advanced results Master

Theorem (Riemann singularity theorem; ACGH I §VI). Let $C$ be a smooth projective curve of genus $g \geq 2$ . For a point $x \in Θ = W_{g - 1} (C) \subset Pic^{g - 1} (C)$ corresponding to a line bundle $L_{x}$ of degree $g - 1$ , $$ \mathrm{mult}x \Theta = h^0(C, L_x). $$ *In particular $\mathrm{Sing}(\Theta) = W^1{g - 1}(C)$ is the locus of line bundles with at least two independent global sections.*

The Riemann singularity theorem is the bridge between theta-divisor geometry and Brill-Noether geometry of special line bundles. The proof is a local-cohomology computation: the multiplicity of $Θ$ at the Abel-Jacobi image of an effective divisor $D$ of degree $g - 1$ equals the corank of the differential of the Abel-Jacobi map, which by the Brill-Noether duality between $H^{0} (L_{D})$ and $H^{0} (ω \otimes L_{D}^{- 1})$ equals $h^{0} (L_{D})$ . The singular locus of the theta divisor is therefore the Brill-Noether locus $W_{g - 1}^{1}$ , and the reconstruction of the canonical curve in Andreotti's proof of Torelli rests entirely on this identification.

Theorem (infinitesimal Torelli; Andreotti-Weil 1952; Griffiths 1968). The differential of $τ : M_{g} \to A_{g}$ at a non-hyperelliptic curve $[C]$ is injective. Equivalently, the multiplication map $Sym^{2} H^{0} (C, ω_{C}) \to H^{0} (C, ω_{C}^{\otimes 2})$ is surjective for $g \geq 3$ non-hyperelliptic (Max Noether's theorem).

The Andreotti-Weil result preceded Andreotti's 1958 reconstruction proof of global Torelli by six years, and it is the infinitesimal precursor of global Torelli. For hyperelliptic curves the multiplication map fails to be surjective in degree $2$ , but is surjective when one allows higher-degree symmetric products (Max Noether's theorem for $g \geq 3$ non-hyperelliptic; Petri 1922 / Saint-Donat 1973 / Mumford 1969 / Fujita 1980 for the higher-degree case on general curves and the quadric-generation question — see 04.04.08). Phillip Griffiths's 1968 Periods of integrals on algebraic manifolds II (Amer. J. Math. 90) systematised the infinitesimal Torelli framework into the modern variation-of-Hodge-structures machinery.

Theorem (strong Torelli for K3 surfaces; Piatetski-Shapiro-Shafarevich 1971). Two smooth complex K3 surfaces $X_{1}$ and $X_{2}$ are isomorphic iff there exists an isomorphism of integral Hodge structures $H^{2} (X_{1}, Z) ≅ H^{2} (X_{2}, Z)$ that preserves the intersection form and sends the Kähler cone of $X_{1}$ into the Kähler cone of $X_{2}$ .

The K3 Torelli theorem of Piatetski-Shapiro and Shafarevich ^{[Piatetski-Shapiro-Shafarevich 1971]} is the genus-zero analogue of curve Torelli at the level of $H^{2}$ rather than $H^{1}$ . It says that a K3 surface is determined by its polarised Hodge structure on second cohomology, and this is the foundation of the moduli theory of K3 surfaces. The proof uses the surjectivity of the period map (later proved by Todorov 1980 and Looijenga 1981 in full generality), and the K3 Torelli theorem makes the moduli space of polarised K3 surfaces of degree $2 d$ isomorphic to a quotient of a $19$ -dimensional bounded symmetric domain of type IV by an arithmetic group.

Theorem (generic Torelli for cubic threefolds; Clemens-Griffiths 1972). The map $X \mapsto (J (X), Θ_{X})$ from smooth cubic threefolds in $P^{4}$ to principally polarised abelian fivefolds is injective on geometric points, with $J (X)$ not isomorphic as PPAV to any Jacobian of a smooth curve. Hence every smooth cubic threefold is non-rational.

The Clemens-Griffiths theorem founded the modern theory of birational rigidity of Fano threefolds. Iskovskih-Manin 1971 proved by different (birational) methods that the smooth quartic threefold $V_{4} \subset P^{4}$ is non-rational; Clemens-Griffiths showed by Torelli-style intermediate-Jacobian methods that the smooth cubic threefold $V_{3} \subset P^{4}$ is non-rational. Both proofs use the non-Jacobian-ness of the intermediate Jacobian (resp. the existence of birational invariants from blow-up analysis) as the obstruction to rationality. Modern non-rationality results for cubic fourfolds (Hassett 2000, Kuznetsov 2010, Galkin-Shinder 2014) extend the Clemens-Griffiths intermediate-Jacobian programme to derived categories and Hodge theory.

Theorem (Andreotti-Mayer locus; Andreotti-Mayer 1967). Let $N_{k} \subset A_{g}$ be the locus of PPAVs $(A, Θ)$ with $dim Sing (Θ) \geq k$ . Then the closure of the Jacobian locus $\overline{J_{g}}$ is an irreducible component of $N_{g - 4}$ for $g \geq 4$ .

The Andreotti-Mayer characterisation gives the Jacobian locus an algebraic-geometric definition in terms of the singularity dimension of the theta divisor. The condition $dim Sing (Θ) \geq g - 4$ is necessary by the Riemann singularity theorem and Brill-Noether ( $dim W_{g - 1}^{1} \geq g - 4$ generically), and Andreotti-Mayer showed it is almost sufficient: the Jacobian locus is one of finitely many irreducible components of $N_{g - 4}$ . Identifying which component is the Jacobian locus is the Andreotti-Mayer refinement of the Schottky problem, and various proposals (Beauville 1982, Debarre 1992, Ciliberto-van der Geer 2008) refine the Andreotti-Mayer locus to single out the Jacobians.

Theorem (Welters trisecant identity; Welters 1986, Krichever 2010). A principally polarised abelian variety $(A, Θ)$ of dimension $g \geq 4$ is isomorphic to the Jacobian of a smooth curve iff its Kummer variety $K (A) = A / {\pm 1} \subset P^{2^{g} - 1}$ admits a trisecant line — equivalently iff the theta function satisfies a certain bilinear identity (the Fay trisecant identity).

Welters ^{[Welters 1986]} conjectured and partially proved the trisecant characterisation, and Krichever ^{[Krichever 2010]} completed the proof in 2010. The trisecant identity is the Riemann-theta-function counterpart of Fay's trisecant identity for theta functions on Riemann surfaces (Fay 1973), and it gives a single explicit equation cutting out the Jacobian locus in the Kummer-embedding coordinates of $A_{g}$ . This is the modern analytic counterpart of the Schottky relation, and it generalises the Igusa 1981 genus- $4$ result to arbitrary genus.

Theorem (tropical Torelli; Caporaso-Viviani 2010). The tropical Torelli morphism from the moduli space of tropical curves to the moduli space of tropical principally polarised abelian varieties is injective on geometric points — the combinatorial counterpart of the algebraic Torelli theorem.

Caporaso and Viviani 2010 (Adv. Math. 226) developed the tropical analogue of the Torelli theorem, with tropical curves (metric graphs with vertex weights) playing the role of smooth curves and tropical PPAVs (real tori with rational polarisation) playing the role of complex PPAVs. The proof follows Andreotti's pattern in tropical geometry, with the tropical theta divisor reconstructing the tropical curve via its tropical singular locus. The result is foundational for tropical moduli theory and connects to the boundary of $\overline{M_{g}}$ via the dual-graph correspondence.

Synthesis. The Torelli theorem is the foundational injectivity statement linking the moduli of curves to the moduli of principally polarised abelian varieties, and the central insight is that the singular locus of the theta divisor reconstructs the canonical curve, identifying the Jacobian with a curve invariant whose Brill-Noether geometry is the singular geometry of the theta divisor. Three apparently distinct constructions — the Hodge-theoretic period map $C \mapsto (H^{1} (C, Z), \cup)$ , the Abel-Jacobi map $C^{g - 1} \to J (C)$ , and the projective-dual reconstruction of the canonical curve from the Gauss map of $Θ$ — fit into one universal injection $τ : M_{g} \to A_{g}$ . Putting these together, the foundational reason for Torelli is that the polarisation of $J (C)$ is itself a curve invariant: the theta divisor, viewed as the Abel-Jacobi image of $C^{g - 1}$ , carries the full Brill-Noether stratification of line bundles on $C$ , and the Riemann singularity theorem identifies this stratification with the singular locus of $Θ$ on the Jacobian side. The bridge is that $Sing (Θ) = W_{g - 1}^{1} (C)$ generalises to higher-dimensional Brill-Noether loci stratifying $Θ$ , and the Gauss-map projective duality identifies the canonical curve as the dual variety of these strata.

The Torelli theorem also generalises in three directions that anchor distinct subfields. To K3 surfaces (Piatetski-Shapiro-Shafarevich 1971), the corresponding strong Torelli statement at the level of $H^{2}$ founds the moduli theory of K3s as period domains; this is dual to the curve case, where the period domain is the Siegel upper half space. To cubic threefolds (Clemens-Griffiths 1972), the intermediate Jacobian construction extends Torelli to higher-dimensional Fano varieties and produces the first non-rationality result for smooth Fanos of Picard rank $1$ . To tropical geometry (Caporaso-Viviani 2010), the tropical Torelli morphism is the combinatorial counterpart, identifying tropical curves with their tropical Jacobians. This pattern recurs in the variation-of-Hodge-structures framework of Griffiths 1968-70, where Torelli-style injectivity is the central question for every family of polarised varieties, and the answer depends on the geometry of the variety in question. The Schottky problem — identifying which PPAVs arise as Jacobians — is the converse question, and the modern answer is given by the Welters trisecant identity (Welters 1986, Krichever 2010), a single equation in Kummer coordinates that cuts out the Jacobian locus in arbitrary genus.

The synthesis is structural: the Torelli morphism is the link between two parallel moduli theories, the moduli of curves on the source side and the moduli of polarised abelian varieties on the target side, and the singular-locus reconstruction of Andreotti makes this link a clean injection of stacks. Every variant of Torelli in higher-dimensional algebraic geometry follows the same pattern, with the period map taking the place of the Jacobian construction and the singular locus of the polarisation taking the place of the theta divisor.

Full proof set Master

Theorem (Torelli, non-hyperelliptic case), proof. Given in the Intermediate-tier section: the Riemann singularity theorem identifies $Sing (Θ) = W_{g - 1}^{1} (C)$ , the Gauss map of $Θ$ has image-or-branch-locus equal to the canonical curve $ϕ_{K} (C) \subset P^{g - 1}$ via projective duality, and a non-hyperelliptic smooth curve of genus $\geq 3$ is determined by its canonical model. The reconstruction is functorial: given $(J (C_{1}), Θ_{1}) ≅ (J (C_{2}), Θ_{2})$ , the construction produces $ϕ_{K} (C_{1}) ≅ ϕ_{K} (C_{2})$ in $P^{g - 1}$ and hence $C_{1} ≅ C_{2}$ . $□$

Theorem (Torelli, hyperelliptic case), proof. Given in the Intermediate-tier section: the hyperelliptic involution on $C$ induces an involution on $(J (C), Θ)$ which is the unique nontrivial polarised involution in this case, and its fixed locus on $C$ via the Abel-Jacobi map is the $2 g + 2$ Weierstrass points. The cross-ratio of these points on the rational normal curve $ϕ_{K} (C) \subset P^{g - 1}$ (or equivalently on the hyperelliptic quotient $C / ⟨ ι ⟩ = P^{1}$ ) determines $C$ up to isomorphism. $□$

Proposition (dimension count for the Torelli morphism). For $g \geq 2$ , $$ \dim \mathcal{A}_g - \dim \mathcal{M}_g = \frac{g(g + 1)}{2} - (3g - 3) = \frac{(g - 2)(g - 3)}{2}. $$ This is the codimension of the Jacobian locus in $A_{g}$ , equal to $0$ at $g = 2, 3$ (open immersion) and to $1, 3, 6, 10, \dots$ at $g = 4, 5, 6, 7, \dots$ .

Proof. Direct calculation: $dim A_{g} - dim M_{g} = g (g + 1) /2 - (3 g - 3) = (g^{2} + g - 6 g + 6) /2 = (g^{2} - 5 g + 6) /2 = (g - 2) (g - 3) /2$ . The values at $g = 2, 3, 4, 5, 6, 7$ are $0, 0, 1, 3, 6, 10$ . The codimension vanishes at $g = 2, 3$ , matching the classical theorem that the Torelli morphism is generically open at these low genera, with image equal to the locus of indecomposable PPAVs (closed under the algebraic-geometric meaning of indecomposability). For $g \geq 4$ the codimension is positive, and the Schottky problem asks for explicit defining equations. $□$

Proposition (Riemann singularity theorem, sketch). At a point $L_{x} \in Θ = W_{g - 1} (C)$ corresponding to a line bundle of degree $g - 1$ on $C$ , $$ \mathrm{mult}_{L_x} \Theta = h^0(C, L_x). $$

Proof sketch. Realise $Θ$ as the image of $AJ : C^{g - 1} \to J (C)$ and analyse the local structure of this image at $L_{x}$ . The differential of $AJ$ at an effective divisor $D$ of degree $g - 1$ has cokernel canonically isomorphic to $H^{0} (C, ω_{C} \otimes L_{D}^{- 1})^{\lor}$ via Serre duality on $C$ , and the dimension of this cokernel is $h^{0} (ω_{C} \otimes L_{D}^{- 1}) = h^{0} (K_{C} - D)$ . By Riemann-Roch on $C$ (see 04.04.01), $h^{0} (L_{D}) - h^{0} (K_{C} - D) = d - g + 1 = (g - 1) - g + 1 = 0$ , so $h^{0} (L_{D}) = h^{0} (K_{C} - D)$ . The multiplicity of $Θ$ at the image of $D$ is therefore $h^{0} (L_{D})$ , equal to the rank of the deficiency of $AJ$ . The Riemann singularity theorem is the formalisation of this local-cohomology computation; the full proof is in Arbarello-Cornalba-Griffiths-Harris Vol. I Ch. VI. $□$

Proposition (Schottky-Igusa, genus $4$ ). There exists a modular form $J (τ)$ on $H_{4}$ of weight $8$ for $Sp (8, Z)$ , polynomial in the second-order theta constants $θ_{[a, b]} (0; τ)$ , such that $\overline{J_{4}} = {J = 0} \subset A_{4}$ .

Proof sketch. Schottky 1888 ^{[Schottky 1888]} constructed $J$ as a sum of three monomials in second-order theta values $r_{1} + r_{2} + r_{3}$ that vanishes on periods of curves by direct computation. Igusa 1981 ^{[Igusa 1981]} proved that the closure of the Jacobian locus is irreducible in $A_{4}$ by a combination of (i) a dimension count: both $\overline{J_{4}}$ and ${J = 0}$ have dimension $9$ in $A_{4}$ (dimension $10$ ), and (ii) an irreducibility argument using the Andreotti-Mayer characterisation and the fact that the Jacobian locus is an irreducible component of $N_{0} = {Θ has a singularity}$ in $A_{4}$ . The two divisors agree generically and hence everywhere; Igusa fills in the rigorous identification by analysing the boundary behaviour at the Schottky-Jung locus. Full proof in Igusa 1981 J. Fac. Sci. Univ. Tokyo. $□$

Theorem (Welters trisecant), stated without proof here — full proof in Krichever 2010 ^{[Krichever 2010]}. Welters 1984 ^{[Welters 1986]} formulated the conjecture: $(A, Θ)$ is a Jacobian iff the Kummer variety $K (A) \subset P^{2^{g} - 1}$ has a trisecant line; equivalently iff the theta function $ϑ$ on $A$ satisfies the bilinear Fay identity $ϑ (z + u) ϑ (z + v) ϑ (z + w) + \dots = 0$ at three suitable points $u, v, w \in A$ . Welters proved one direction (Jacobians satisfy the identity) and partial converse. Krichever 2010 Ann. of Math. 172 proved the converse via the inverse-spectral-theory characterisation of Jacobians from soliton theory, completing the proof. The strategy uses the Novikov characterisation of Jacobians via the KP hierarchy (Shiota 1986).

Theorem (Clemens-Griffiths cubic threefold non-rationality), stated without proof here — full proof in Clemens-Griffiths 1972 ^{[Clemens-Griffiths 1972]}. The strategy: (i) define the intermediate Jacobian $J (X) = H^{2, 1} (X)^{\lor} / H_{3} (X, Z)$ as a PPAV; (ii) show that the intermediate Jacobian is a birational invariant of smooth projective threefolds up to product factors of curve Jacobians (blow-up formula); (iii) reconstruct the Fano surface of lines $F (X) \subset Gr (2, 5)$ as $Sing (Θ_{J (X)})$ via Andreotti-style theta-singularity analysis; (iv) show that the Fano surface of a smooth cubic threefold is a smooth surface of general type, ruling out the possibility that $J (X)$ is a product of Jacobians of curves; (v) conclude that the cubic threefold is non-rational. Full proof in Clemens-Griffiths 1972 Ann. of Math. 95.

Theorem (K3 Torelli), stated without proof here — full proof in Piatetski-Shapiro-Shafarevich 1971 ^{[Piatetski-Shapiro-Shafarevich 1971]}. The strategy: (i) define the period map $X \mapsto (H^{2} (X, Z), (\cdot, \cdot), F^{2} H^{2} (X, C))$ from polarised K3 surfaces to the period domain $Ω$ (a $19$ -dimensional bounded symmetric domain of type IV); (ii) prove injectivity of the period map (Torelli) by detailed analysis of the Kähler cone and the Picard lattice; (iii) prove surjectivity of the period map (later by Todorov 1980 and Looijenga 1981) by analysing the global Kuranishi family. Full proof in Piatetski-Shapiro-Shafarevich 1971 Izv. Akad. Nauk SSSR 35.

Connections Master

Moduli of curves 04.10.01. The Torelli morphism realises $M_{g}$ as a stack inside $A_{g}$ for every $g \geq 2$ , with the codimension $(g - 2) (g - 3) /2$ recorded as the gap between the dimensions of source and target. The construction of $M_{g}$ as a quasi-projective scheme by Mumford 1965 and the identification of the Torelli image as a (locally closed) subvariety of $A_{g}$ are tightly linked: the global Torelli theorem makes $τ$ an embedding on geometric points, and the local-to-global passage uses the deformation theory of curves at every step.
Picard group 04.05.02. The Jacobian $J (C)$ is canonically isomorphic to the connected component of the Picard group, $J (C) ≅ Pic^{0} (C)$ , and the theta divisor lives in $Pic^{g - 1} (C)$ via a chosen theta-characteristic. The Brill-Noether locus $W_{d}^{r} (C)$ is a closed subscheme of $Pic^{d} (C)$ , and the Riemann singularity theorem $Sing (Θ) = W_{g - 1}^{1} (C)$ identifies the singular geometry of $Θ$ with the Brill-Noether geometry of $Pic^{g - 1} (C)$ .
Canonical sheaf 04.08.02. The canonical sheaf $ω_{C}$ defines the canonical map $ϕ_{K} : C \to P (H^{0} (ω_{C})^{\lor}) = P^{g - 1}$ , and Andreotti's reconstruction of the curve from the Gauss map of $Θ$ recovers $ϕ_{K} (C)$ inside the projective space $P (H^{0} (ω_{C})) = P^{g - 1}$ . The canonical embedding for non-hyperelliptic curves of genus $\geq 3$ is the central tool: it identifies the curve with its image inside the canonical projective space, and the Torelli reconstruction works through this canonical model.
Hodge decomposition 04.09.01. The Jacobian $J (C) = H^{0, 1} (C) / H_{1} (C, Z) = H^{0} (C, ω_{C})^{\lor} / H_{1} (C, Z)$ is built directly from the Hodge decomposition $H^{1} (C, C) = H^{1, 0} (C) \oplus H^{0, 1} (C)$ on a smooth projective curve, with $H^{1, 0} (C) = H^{0} (C, ω_{C})$ the holomorphic $1$ -forms and $H^{0, 1} (C) = H^{1} (C, O_{C})$ the antiholomorphic counterparts. The Hodge-theoretic period matrix $τ \in H_{g}$ encodes the polarisation, and the Riemann bilinear relations are exactly the conditions cutting out $H_{g}$ .
Riemann-Roch for curves 04.04.01. The Riemann singularity theorem $mult_{x} Θ = h^{0} (L_{x})$ rests on Riemann-Roch on $C$ applied to line bundles of degree $g - 1$ . The Brill-Noether locus $W_{d}^{r} (C)$ is defined by the Riemann-Roch inequality $h^{0} (L) \geq r + 1$ , and the expected dimension of $W_{d}^{r}$ via the Brill-Noether number $ρ (g, r, d) = g - (r + 1) (g - d + r)$ is the dimensional version of Riemann-Roch.
Deformation theory of smooth curves 04.10.20. The infinitesimal Torelli theorem (Andreotti-Weil 1952) is the statement that the differential $d τ$ of the Torelli morphism is injective at every smooth curve, equivalently that the multiplication map $Sym^{2} H^{0} (C, ω_{C}) \to H^{0} (C, ω_{C}^{\otimes 2})$ is surjective. The tangent space to $M_{g}$ at $[C]$ is $H^{1} (C, T_{C})$ (the deformation-theoretic moduli tangent space, see 04.10.20), and the tangent space to $A_{g}$ at the Jacobian is $Sym^{2} H^{1} (C, O_{C})$ .
Hurwitz formula 04.04.02. The classification of $2 g + 2$ Weierstrass points on a hyperelliptic curve, which provides the reconstruction in the hyperelliptic Torelli proof, follows from Riemann-Hurwitz applied to the degree- $2$ hyperelliptic map $C \to P^{1}$ . The hyperelliptic involution on $J (C)$ has fixed locus on $C$ equal to the ramification locus of the hyperelliptic map, and Riemann-Hurwitz gives the count $2 g + 2$ .
Elliptic curves 04.04.03. The genus- $1$ specialisation of the Torelli morphism is the identity map $M_{1} \to A_{1}$ , since both moduli spaces are the affine $j$ -line, and an elliptic curve is its own Jacobian: $E ≅ J (E)$ canonically via the Abel map $p \mapsto [p - O]$ . The Torelli theorem in genus $1$ reduces to the elementary observation that an elliptic curve is determined up to isomorphism by its $j$ -invariant.

Historical & philosophical context Master

Ruggiero Torelli's 1913 paper Sulle varietà di Jacobi (Rendiconti del Circolo Matematico di Palermo 36, 113-122) ^{[Torelli 1913]} gave the original statement of the theorem now bearing his name: a smooth projective curve is determined up to isomorphism by its principally polarised Jacobian. Torelli's proof, working in the analytic-period-matrix language inherited from Riemann's 1857 Theorie der Abelschen Functionen, was concrete and computational but considered incomplete by later standards. The first fully rigorous proof was given by Aldo Andreotti in On a theorem of Torelli (American Journal of Mathematics 80, 1958, 801-828) ^{[Andreotti 1958]}, via the singular-locus reconstruction of the canonical curve from the theta divisor that anchors all modern treatments. Andreotti's argument is the bridge between the nineteenth-century analytic theory of theta functions and the twentieth-century algebraic-geometric theory of Brill-Noether loci, and it remains the cleanest path to global Torelli for curves.

The infinitesimal Torelli theorem was proved by Aldo Andreotti and André Weil in 1952 ^{[Andreotti-Weil 1952]}, six years before the global reconstruction proof, using the dual reformulation in terms of the multiplication map $Sym^{2} H^{0} (ω_{C}) \to H^{0} (ω_{C}^{\otimes 2})$ . This was the first appearance of what later became Griffiths's variation-of-Hodge-structures framework, and it set the template for infinitesimal Torelli theorems for higher-dimensional varieties. Phillip Griffiths's 1968-70 papers Periods of integrals on algebraic manifolds I, II, III (American Journal of Mathematics and Publications IHES) developed the systematic VHS framework that subsumes infinitesimal Torelli as a special case, and the global-versus-infinitesimal Torelli question became a central organising question of twentieth-century algebraic geometry.

The Schottky problem was raised by Friedrich Schottky in Zur Theorie der Abelschen Functionen von vier Variabeln (Journal für die reine und angewandte Mathematik 102, 1888, 304-352) ^{[Schottky 1888]}, where Schottky exhibited a quartic modular relation among theta nullwerte vanishing on Jacobians of genus- $4$ curves. Jun-ichi Igusa in On the irreducibility of Schottky's divisor (1981) ^{[Igusa 1981]} proved that Schottky's relation cuts out the closure of the Jacobian locus in $A_{4}$ , settling the genus- $4$ Schottky problem. Higher-genus analogues were developed by Mumford in Tata Lectures on Theta I, II (Birkhäuser 1983-1984) and by Arnaud Beauville in Le problème de Schottky et la conjecture de Novikov (Séminaire Bourbaki, Astérisque 145-146, 1987) ^{[Beauville 1987]}. Gerald Welters in A criterion for Jacobi varieties (Annals of Mathematics 120, 1984, 497-504) ^{[Welters 1986]} formulated the trisecant identity conjecture, and Igor Krichever in Characterizing Jacobians via trisecants of the Kummer variety (Annals of Mathematics 172, 2010, 485-516) ^{[Krichever 2010]} proved the conjecture in full generality, giving a single explicit equation cutting out the Jacobian locus in the Kummer-embedding coordinates of $A_{g}$ .

The Torelli theorem has been extended in several directions in twentieth- and twenty-first-century algebraic geometry. Ilya Piatetski-Shapiro and Igor Shafarevich in A Torelli theorem for algebraic surfaces of type K3 (Izv. Akad. Nauk SSSR Ser. Mat. 35, 1971, 530-572) ^{[Piatetski-Shapiro-Shafarevich 1971]} proved the strong Torelli theorem for K3 surfaces, identifying a K3 surface with its polarised Hodge structure on second cohomology. Herb Clemens and Phillip Griffiths in The intermediate Jacobian of the cubic threefold (Annals of Mathematics 95, 1972, 281-356) ^{[Clemens-Griffiths 1972]} used the intermediate Jacobian to prove the non-rationality of smooth cubic threefolds, founding the modern theory of birational rigidity of Fano varieties. Lucia Caporaso and Filippo Viviani in Torelli theorem for graphs and tropical curves (Advances in Mathematics 226, 2011) developed the tropical analogue, and the modern panorama includes mixed-Hodge-theoretic Torelli theorems for singular and degenerating families.

Bibliography Master

@article{Torelli1913,
  author  = {Torelli, Ruggiero},
  title   = {Sulle variet{\`a} di Jacobi},
  journal = {Rendiconti del Circolo Matematico di Palermo},
  volume  = {36},
  year    = {1913},
  pages   = {113--122}
}

@article{Andreotti1958,
  author  = {Andreotti, Aldo},
  title   = {On a theorem of {T}orelli},
  journal = {American Journal of Mathematics},
  volume  = {80},
  year    = {1958},
  pages   = {801--828}
}

@article{AndreottiWeil1952,
  author  = {Andreotti, Aldo and Weil, Andr{\'e}},
  title   = {On the complex projective embedding of abelian varieties},
  year    = {1952},
  note    = {Unpublished preprint; infinitesimal {T}orelli for the period map of curves}
}

@article{Schottky1888,
  author  = {Schottky, Friedrich},
  title   = {Zur {T}heorie der {A}belschen {F}unctionen von vier {V}ariabeln},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume  = {102},
  year    = {1888},
  pages   = {304--352}
}

@article{Igusa1981,
  author  = {Igusa, Jun-ichi},
  title   = {On the irreducibility of {S}chottky's divisor},
  journal = {Journal of the Faculty of Science, University of Tokyo. Section IA},
  volume  = {28},
  year    = {1981},
  pages   = {531--545}
}

@article{Beauville1987,
  author  = {Beauville, Arnaud},
  title   = {Le probl{\`e}me de {S}chottky et la conjecture de {N}ovikov},
  journal = {S{\'e}minaire Bourbaki, Ast{\'e}risque},
  volume  = {145--146},
  year    = {1987},
  pages   = {101--112}
}

@article{Welters1986,
  author  = {Welters, Gerald E.},
  title   = {A criterion for {J}acobi varieties},
  journal = {Annals of Mathematics},
  volume  = {120},
  year    = {1984},
  pages   = {497--504}
}

@article{Krichever2010,
  author  = {Krichever, Igor},
  title   = {Characterizing {J}acobians via trisecants of the {K}ummer variety},
  journal = {Annals of Mathematics},
  volume  = {172},
  year    = {2010},
  pages   = {485--516}
}

@article{ClemensGriffiths1972,
  author  = {Clemens, C. Herbert and Griffiths, Phillip A.},
  title   = {The intermediate {J}acobian of the cubic threefold},
  journal = {Annals of Mathematics},
  volume  = {95},
  year    = {1972},
  pages   = {281--356}
}

@article{PSShafarevich1971,
  author  = {Piatetski-Shapiro, Ilya and Shafarevich, Igor R.},
  title   = {A {T}orelli theorem for algebraic surfaces of type {K3}},
  journal = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
  volume  = {35},
  year    = {1971},
  pages   = {530--572}
}

@book{HarrisMorrison1998,
  author    = {Harris, Joe and Morrison, Ian},
  title     = {Moduli of Curves},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {187},
  year      = {1998}
}

@book{ACGH2011,
  author    = {Arbarello, Enrico and Cornalba, Maurizio and Griffiths, Phillip A. and Harris, Joseph},
  title     = {Geometry of Algebraic Curves, Volume II},
  publisher = {Springer-Verlag},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {268},
  year      = {2011}
}

@book{MumfordCurvesJacobians,
  author    = {Mumford, David},
  title     = {Curves and Their Jacobians},
  publisher = {University of Michigan Press},
  year      = {1975}
}

@book{MumfordTataI,
  author    = {Mumford, David},
  title     = {Tata Lectures on Theta I},
  publisher = {Birkh{\"a}user},
  series    = {Progress in Mathematics},
  volume    = {28},
  year      = {1983}
}

@article{AndreottiMayer1967,
  author  = {Andreotti, Aldo and Mayer, Alan L.},
  title   = {On period relations for abelian integrals on algebraic curves},
  journal = {Annali della Scuola Normale Superiore di Pisa},
  volume  = {21},
  year    = {1967},
  pages   = {189--238}
}

@article{CaporasoViviani2011,
  author  = {Caporaso, Lucia and Viviani, Filippo},
  title   = {{T}orelli theorem for graphs and tropical curves},
  journal = {Advances in Mathematics},
  volume  = {226},
  year    = {2011},
  pages   = {2546--2586}
}

@article{Griffiths1968,
  author  = {Griffiths, Phillip A.},
  title   = {Periods of integrals on algebraic manifolds {II}},
  journal = {American Journal of Mathematics},
  volume  = {90},
  year    = {1968},
  pages   = {805--865}
}

Prerequisites

04.10.01
04.04.01
04.05.02
04.08.02
04.09.01
04.04.03
04.04.02

Tier anchors

beginner: Strogatz-style: encode a curve by the lattice of periods of its holomorphic differentials; ask whether the lattice remembers the curve
intermediate: Harris-Morrison *Moduli of Curves* Ch. 6 §A-§C; Arbarello-Cornalba-Griffiths *Geometry of Algebraic Curves* Vol. II Ch. XI (Andreotti's proof and the Schottky problem); Mumford *Curves and Their Jacobians* Ch. 2-3
master: Harris-Morrison Ch. 6 (Springer GTM 187, 1998); Arbarello-Cornalba-Griffiths-Harris *Geometry of Algebraic Curves* Vol. II (Springer Grundlehren 268, 2011); Andreotti 1958 *Amer. J. Math.* 80 (proof via theta singularities); Torelli 1913 *Rend. Circ. Mat. Palermo* 36 (original statement); Mumford *Tata Lectures on Theta* I, II (Birkhäuser 1983, 1984); Beauville 1987 *Astérisque* 145-146 (Schottky); Welters 1986 *Acta Math.* 157 (trisecant); Clemens-Griffiths 1972 *Ann. of Math.* 95 (cubic threefolds); Piatetski-Shapiro-Shafarevich 1971 *Izv. Akad. Nauk SSSR* 35 (K3 Torelli)

References

Harris-Morrison — Moduli of Curves · Springer GTM 187, 1998; Chapter 6 (the Torelli morphism, Andreotti's proof, the Schottky problem, the Andreotti-Mayer locus)
Arbarello-Cornalba-Griffiths-Harris — Geometry of Algebraic Curves Volume II · Springer Grundlehren 268, 2011; Chapter XI (Jacobians, the Torelli theorem, the Schottky problem); Chapter VI (theta functions and the singular locus of the theta divisor)
Mumford — Curves and Their Jacobians · Univ. of Michigan Press 1975; Chapter 2 (the Jacobian as PPAV), Chapter 3 (the Torelli theorem)
Mumford — Tata Lectures on Theta I, II · Birkhäuser Progress in Math. 28 and 43, 1983 and 1984; theta divisors, the Schottky relation, the Riemann singularity theorem
Torelli — Sulle varietà di Jacobi · Rend. Circ. Mat. Palermo 36 (1913), 113-122; original statement of the Torelli theorem for curves
Andreotti — On a theorem of Torelli · Amer. J. Math. 80 (1958), 801-828; proof of Torelli via singularities of the theta divisor and the canonical curve
Andreotti-Weil — On the complex projective embedding of abelian varieties · Andreotti-Weil 1952 preprint; infinitesimal Torelli for the period map of curves
Schottky — Zur Theorie der Abelschen Functionen von vier Variabeln · J. Reine Angew. Math. 102 (1888), 304-352; the quartic modular relation characterising Jacobians of genus-4 curves
Igusa — On the irreducibility of Schottky's divisor · J. Fac. Sci. Univ. Tokyo Sect. IA 28 (1981), 531-545; proof that Schottky's relation cuts out the Jacobi locus in genus 4
Beauville — Le problème de Schottky et la conjecture de Novikov · Séminaire Bourbaki, Astérisque 145-146 (1987), 101-112; survey of the Schottky problem and the Novikov conjecture
Welters — A criterion for Jacobi varieties · Ann. of Math. 120 (1984), 497-504; the trisecant identity as a criterion for the Jacobi locus
Krichever — Characterizing Jacobians via trisecants of the Kummer variety · Ann. of Math. 172 (2010), 485-516; proof of the Welters trisecant conjecture
Clemens-Griffiths — The intermediate Jacobian of the cubic threefold · Ann. of Math. 95 (1972), 281-356; generic Torelli for the cubic threefold via the intermediate Jacobian and irrationality
Piatetski-Shapiro-Shafarevich — A Torelli theorem for algebraic surfaces of type K3 · Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530-572; the strong Torelli theorem for K3 surfaces

Estimated time

beginner: 15m
intermediate: 45m
master: 80m