Petri map mu_0 and Gieseker-Petri theorem

Intuition [Beginner]

A smooth projective curve $C$ of genus $g$ carries line bundles — bundles of "sections" that assign a function to each point, possibly with poles. A linear series is a subspace of these sections. The central question: for which numerical parameters $(d,r)$ does a curve carry a linear series of degree $d$ and dimension $r$?

The Petri map is a single linear-algebra object that answers this question. Given a line bundle $L$ on $C$, the Petri map multiplies a section $s$ of $L$ by a section $t$ of the "complementary" bundle $K_C(-L)$, producing a differential form $s\cdot t$. This multiplication map is the engine behind the Brill-Noether theory of linear series on curves.

When this multiplication map is injective — no non-zero pair $(s,t)$ produces the zero form — the curve is "generic" in the sense of having the expected number of linear series. Gieseker proved in 1982 that on a general curve of genus $g$, the Petri map is injective for every line bundle $L$.

Visual [Beginner]

A curve $C$ with two vector spaces drawn above it: $H^0(C,L)$ on the left and $H^0(C,K_C(-L))$ on the right. An arrow labelled ${\mu }_0$ goes from their product space to the space of differential forms $H^0(C,K_C)$. The kernel of ${\mu }_0$ is shown as the "obstruction" — the part of the product space that maps to zero.

The dimension of the kernel controls exactly how many "unexpected" linear series the curve carries.

Worked example [Beginner]

Take a general curve $C$ of genus $g=4$. By Brill-Noether theory, $C$ is expected to carry a $g_3^1$ — a degree-3 linear series of dimension 1 — when the Brill-Noether number $\rho =g-2(g-d+r)=4-2(4-3+1)=4-4=0$.

For this $L=g_3^1$, the Petri map is ${\mu }_0:H^0(L)\times H^0(K_C(-L))\to H^0(K_C)$. Here $h^0(L)=2$ and $h^0(K_C(-L))=h^0(K_C)-3+1-g+1$ by Riemann-Roch; with $g=4$ and $\mathrm{deg}(K_C(-L))=2g-2-3=3$, Riemann-Roch gives $h^0(K_C(-L))=3-4+1+h^0(L)=h^0(L)=2$ (since $\mathrm{deg}L=3>2g-2=6$ is false, so we use Serre duality instead).

The Petri map goes from a $2\times 2=4$-dimensional tensor product to $H^0(K_C)$ which is $g=4$ dimensional. Injectivity means this $4\to 4$ map is an isomorphism, and for a general genus-4 curve it is.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $C$ be a smooth projective curve of genus $g$ over an algebraically closed field $k$, and let $L$ be a line bundle on $C$.

Definition (Petri map). The Petri map (or Petri homomorphism) associated to $L$ is the $k$-linear map $$ \mu_0(L) : H^0(C, L) \otimes_k H^0(C, K_C \otimes L^{-1}) \longrightarrow H^0(C, K_C) $$ defined on pure tensors by ${\mu }_0(s\otimes t)=s\cdot t$, the product of the section $s$ of $L$ with the section $t$ of $K_C\otimes L^{-1}$, landing in $H^0(C,K_C)$ via the natural inclusion $L\otimes (K_C\otimes L^{-1})\hookrightarrow K_C$. The map extends to all tensors by $k$-linearity.

The Petri map is natural in $L$: given a morphism of line bundles $L\to L^{\prime }$, there is a commutative diagram relating ${\mu }_0(L)$ and ${\mu }_0(L^{\prime })$ ^{[Arbarello-Cornalba-Griffiths-Harris Ch. I]}.

Definition (Petri condition). A curve $C$ satisfies the Petri condition if for every line bundle $L$ on $C$, the Petri map ${\mu }_0(L)$ is injective.

Deformation-theoretic interpretation. The kernel $\ker {\mu }_0(L)$ is identified with the tangent space to the Brill-Noether locus $W_d^r(C)=\{L\in {\mathrm{Pic}}^d(C):h^0(C,L)\ge r+1\}$ at the point $[L]$, where $r=h^0(C,L)-1$ and $d=\mathrm{deg}L$. Specifically, ^{[Arbarello-Cornalba-Griffiths-Harris Ch. IV]} $$ T_{[L]} W^r_d(C) \cong (\ker \mu_0(L))^\perp \subseteq H^1(C, \mathcal{O}C), $$ where the orthogonal complement is taken with respect to the Serre duality pairing. Injectivity of ${\mu }_0$ therefore means $\dim T{[L]} W^r_d = \rho = g - (r+1)(g - d + r)$, the expected dimension.

Counterexamples to common slips.

• Petri condition fails on special curves. On a hyperelliptic curve of genus $g\ge 2$, the $g_2^1$ (the hyperelliptic pencil) has Petri map that is not injective: the two sections of the pencil multiply against the $g-2$ independent sections of $K_C\otimes L^{-1}$ but satisfy a relation. The kernel has dimension $g-2$.
• Confusing ${\mu }_0$ with the full multiplication map. The Petri map is not $H^0(L)\otimes H^0(L)\to H^0(L^2)$. The correct complementary bundle is $K_C\otimes L^{-1}$, not $L$ itself.
• Injectivity for all $L$ simultaneously. The Petri condition requires injectivity for every line bundle $L$ on $C$, not just one.
• Characteristic issues. In positive characteristic, the Petri map can fail to be injective for reasons that do not arise in characteristic zero. Kempf's proof of the Gieseker-Petri theorem uses characteristic-zero methods.

Key theorem with proof [Intermediate+]

Theorem (Gieseker-Petri). Let $C$ be a general smooth projective curve of genus $g$ over an algebraically closed field of characteristic zero. Then $C$ satisfies the Petri condition: for every line bundle $L$ on $C$, the Petri map ${\mu }_0(L)$ is injective.

Proof (sketch, following Eisenbud-Harris 1983). The argument proceeds by degeneration. Fix a generic $(g+1)$-pointed stable curve $X_0$ of genus $g$ consisting of $g$ rational components $C_1,\dots ,C_g$ arranged in a chain, each meeting the next at a single node, with $C_i\cong {\mathbb{P}}^1$ carrying two marked points $p_i,q_i$ in addition to the nodes.

Step 1: Limit linear series. Eisenbud-Harris develop the theory of limit linear series on such a chain curve ^{[Eisenbud-Harris 1983]}. A limit $g_d^r$ on $X_0$ consists of a linear series ${\ell }_i$ of degree $d$ and dimension $r$ on each component $C_i\cong {\mathbb{P}}^1$, satisfying compatibility conditions at the nodes (the vanishing sequence at the node on one side matches the adjusted vanishing sequence on the adjacent component).

Step 2: Petri map on a rational component. On $C_i\cong {\mathbb{P}}^1$, every line bundle is $\mathcal{O}(d_i)$ for some $d_i$. The Petri map for $\mathcal{O}(d_i)$ on ${\mathbb{P}}^1$ reduces to the multiplication map $k[x{]}_{\le a}\otimes k[x{]}_{\le b}\to k[x{]}_{\le a+b}$ which is always injective (polynomial rings have no zero divisors). Hence the Petri map is injective on each component.

Step 3: Compatibility forces injectivity on the chain. A non-zero element of $\ker {\mu }_0$ on the chain curve would restrict to a non-zero element of the kernel on some component, contradicting Step 2. The compatibility conditions at the nodes, combined with the injectivity on each component, force the kernel on the entire chain to be zero.

Step 4: Semistable reduction and the general curve. Let $\mathcal{C}\to B$ be the family of curves over the moduli space ${\overline{\mathcal{M}}}_g$ with special fibre the chain curve $X_0$. A limit linear series on $X_0$ with injective Petri map smooths to a linear series on the general fibre with injective Petri map, by upper-semicontinuity of $h^0$ and the deformation theory of the Petri map. Since the chain curve has no degenerate limit linear series (Step 3), the general curve has no degenerate linear series either. $\square$

Bridge. The Petri map encodes the tangent-space dimension of the Brill-Noether locus $W_d^r$ inside ${\mathrm{Pic}}^d(C)$, and its injectivity for a general curve is the geometric engine behind three interconnected results: the dimension formula $\dim W_d^r=\rho$ from Riemann-Roch 04.04.01, the smoothness of $W_d^r$ at points where the Petri map is injective (via the deformation-theoretic identification using the canonical sheaf 04.08.02 and Serre duality 04.08.03), and the existence theorem for linear series of expected dimension $\rho \ge 0$. The same multiplication map reappears in Clifford's theorem 04.04.09 as the algebraic constraint that bounds $h^0(L)$ above when $L$ is special, and in Lazarsfeld's proof 04.04.16 where injectivity of ${\mu }_0$ on a K3 surface's hyperplane section yields the Brill-Noether-Petri theorem without degeneration.

Exercises [Intermediate+]

Advanced results [Master]

Brill-Noether dimension theorem. The Gieseker-Petri theorem implies the dimension formula for Brill-Noether loci: for a general curve $C$ of genus $g$, the locus $W_d^r(C)\subset {\mathrm{Pic}}^d(C)$ is either empty (if $\rho <0$) or has pure dimension $\rho =g-(r+1)(g-d+r)$ (if $\rho \ge 0$), and is smooth away from $W_d^{r+1}(C)$. The proof identifies $T_{[L]}{W}_d^r=(\ker {\mu }_0(L){)}^{\perp }$ via the deformation theory of linear series; injectivity of ${\mu }_0$ gives $\dim T_{[L]}{W}_d^r=\rho$, and smoothness follows.

Griffiths-Harris dimension theorem (alternate proof). Griffiths and Harris (1980) gave an independent proof of the Brill-Noether dimension theorem via a Schubert-cycle computation on the symmetric product $C_d$. The dimension of $W_d^r(C)$ is computed by the class $c_1(\mathcal{O}(1){)}^{...}$ in the cohomology ring of the Grassmannian $G(r+1,d-g+r+1)$ pulled back to $C_d$ via the Abel-Jacobi map. The Gieseker-Petri theorem strengthens this by proving smoothness, not just dimension.

Base-point-free pencil trick. A key computational tool: if $L$ is base-point-free with $h^0(L)=2$ (a pencil), the evaluation sequence $0\to H^0(L{)}^{\vee }\otimes H^0(K_C\otimes L^{-1})\to H^0(K_C)\to H^0(K_C\otimes L^{-1}{|}_D)\to 0$ is exact where $D$ is the zero-locus of a general section of $L$. This identifies the image of ${\mu }_0$ and shows $\dim \ker {\mu }_0=g-2r-2+\rho$ for a $g_d^r$.

Kempf's proof of Gieseker-Petri. Kempf (1980) gave a characteristic-free proof of the Gieseker-Petri theorem in the case $\rho =0$, using the determinantal structure of the Petri map. The Petri matrix is expressed as a Jacobian matrix of explicit theta functions on the Jacobian, and the non-vanishing of the determinant follows from Riemann's theta singularity theorem.

Lazarsfeld's proof (1994). Lazarsfeld gave a proof of the Gieseker-Petri theorem without degeneration, using vector bundles on K3 surfaces. If $S$ is a K3 surface and $C\subset S$ is a smooth curve of genus $g$ in the primitive class $[C]$, and $L$ is any line bundle on $C$, then the Petri map ${\mu }_0(L)$ on $C$ is injective. This uses the restriction theorem for vector bundles on K3 surfaces and is covered in unit 04.04.16.

Synthesis. The Petri map is the cohomological fingerprint of a curve's linear series, encoding three interconnected geometric phenomena. The first is the multiplication map ${\mu }_0:H^0(L)\otimes H^0(K_C\otimes L^{-1})\to H^0(K_C)$ whose kernel measures the failure of a curve to be "general" in the Brill-Noether sense 04.04.01. The second is the tangent-space identification $\ker {\mu }_0\cong T_{[L]}{W}_d^r$ connecting cohomology to deformation theory via the canonical sheaf 04.08.02 and Serre duality 04.08.03. The third is the degeneration strategy of Gieseker and Eisenbud-Harris, replacing the general curve with a chain of rational curves where injectivity reduces to polynomial algebra, then lifting back via semistable reduction — a technique that reappears in the proof of Martens' theorem 04.04.10 and the Fulton-Lazarsfeld connectedness theorem 04.04.15. These three perspectives — cohomological, deformation-theoretic, and degeneration-theoretic — are unified by the single linear-algebra object ${\mu }_0$, and the Gieseker-Petri theorem is the statement that this object is as well-behaved as possible for a general curve.

Full proof set [Master]

The full proof of the Gieseker-Petri theorem via the Eisenbud-Harris limit-linear-series method is sketched in the Key theorem section. The Kempf proof for $\rho =0$ proceeds as follows: identify the Petri matrix with the matrix of partial derivatives of the theta function $\theta$ on the Jacobian $J(C)$ at the point corresponding to the line bundle $L$; Riemann's singularity theorem identifies the multiplicity of $\theta$ at this point with $h^0(L)$; the Petri matrix is the Hessian of $\theta$ at a singular point; non-degeneracy follows from the fact that the theta divisor is an ample divisor on $J(C)$ and its singular locus has the expected codimension. The Lazarsfeld proof is deferred to unit 04.04.16.

Connections [Master]

• Riemann-Roch theorem 04.04.01 — the Petri map's domain dimension is computed by the Riemann-Roch formula applied to both $L$ and $K_C\otimes L^{-1}$; the Brill-Noether number $\rho =g-(r+1)(g-d+r)$ is a direct descendant of Riemann-Roch.

• Canonical sheaf 04.08.02 — the codomain of ${\mu }_0$ is $H^0(C,K_C)$, the space of global differentials; the Petri map is the multiplication pairing between sections of $L$ and sections of the complementary bundle $K_C\otimes L^{-1}$.

• Serre duality 04.08.03 — the identification $H^0(C,K_C\otimes L^{-1})\cong H^1(C,L{)}^{\vee }$ via Serre duality makes the Petri map the dual of the coboundary map in the long exact cohomology sequence of $0\to L^{-1}\to {\mathcal{O}}_C\to {\mathcal{O}}_D\to 0$ for an effective divisor $D$ with $\mathcal{O}(D)=L$.

• Clifford's theorem 04.04.09 — Clifford's inequality $h^0(L)\le d/2+1$ for special line bundles is proved using a kernel element of the Petri map for $L=\mathcal{O}(D)$ with $D$ an effective special divisor.

• Gonality 04.04.11 — the Petri map for the $g_k^1$ (the gonality map) controls the existence and uniqueness of the minimal-degree map to ${\mathbb{P}}^1$.

• Lazarsfeld's K3 proof 04.04.16 — a proof of the Petri theorem using vector bundles on K3 surfaces and the restriction theorem, avoiding degeneration entirely.

Historical & philosophical context [Master]

Karl Petri stated and proved the injectivity result for the multiplication map now bearing his name in the 1922 paper Über Spezialkurven (Math. Ann. 85, 27-52) ^{[Petri 1922]}. Petri's proof contained a gap in the base-point-free pencil trick, which was corrected by several authors over the subsequent decades. The result became a central pillar of Brill-Noether theory after the modern reformulation by Arbarello, Cornalba, Griffiths, and Harris in Geometry of Algebraic Curves Vol. I (1985).

David Gieseker gave the first complete proof of the general statement in 1982 ^{[Gieseker 1982]}, using degeneration to stable curves and showing that the Petri map is injective for a general curve. Gieseker's proof was technically demanding; Eisenbud and Harris (1983) simplified it enormously with their theory of limit linear series on reducible curves ^{[Eisenbud-Harris 1983]}, reducing the injectivity check to polynomial multiplication on rational components.

George Kempf (1980) independently proved the $\rho =0$ case via theta-function methods, connecting the Petri matrix to the Hessian of the Riemann theta function at a singularity. Lazarsfeld (1994) gave a fourth proof using vector bundles on K3 surfaces, which avoids degeneration entirely and provides the most conceptual explanation for why the Petri map is injective.

The philosophical content: a "general" curve is the most common type of curve, and the Petri condition describes the generic behaviour of linear series. Curves violating the Petri condition (hyperelliptic, trigonal, plane quintics, etc.) form proper subvarieties of the moduli space ${\mathcal{M}}_g$.

Bibliography [Master]

• Petri, K. (1922). Über Spezialkurren. Mathematische Annalen 85, 27-52. [Originator paper.]
• Gieseker, D. (1982). Stable curves and special divisors: Petri's conjecture. Inventiones Mathematicae 71, 33-53. [First complete proof.]
• Eisenbud, D.; Harris, J. (1983). Divisors on general curves. Inventiones Mathematicae 74, 271-290. [Limit linear series proof.]
• Kempf, G. (1980). Schubert methods with an application to algebraic curves. Publ. Math. Centrum, Amsterdam. [Theta-function proof for $\rho =0$.]
• Lazarsfeld, R. (1994). Brill-Noether-Petri without degenerations. Unpublished lecture; see ACGH Ch. VIII for an exposition. [K3-surface proof.]
• Arbarello, E.; Cornalba, M.; Griffiths, P.; Harris, J. Geometry of Algebraic Curves, Vol. I. Grundlehren 267. [Ch. I and Ch. IV; standard reference.]
• Hartshorne, R. Algebraic Geometry. Springer GTM 52, §IV.1. [Linear series on curves.]