04.04.15 · algebraic-geometry / curves

Fulton-Lazarsfeld connectedness theorem

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Anchor (Master): Fulton-Lazarsfeld 1981 *Positive polynomials for ample vector bundles* (Ann. Math. 118, originator); Fulton — Intersection Theory Ch. 12; Lazarsfeld — Positivity in Algebraic Geometry I

Intuition [Beginner]

The Fulton-Lazarsfeld connectedness theorem says: if you have a map between two "positive" (ample) vector bundles on an irreducible variety, and you look at the set where the map drops rank by the maximum possible amount, then that set is connected.

Why is this surprising? The degeneracy locus is cut out by many equations (all the minors of a matrix), and in general, the intersection of many hypersurfaces can have many disconnected components. The Fulton-Lazarsfeld theorem says that when the bundles are ample, the geometry is rigid enough to force connectedness.

The simplest instance: the intersection of two ample divisors on a projective variety is connected. This is the classical Bertini-type result. Fulton-Lazarsfeld generalises this to degeneracy loci of arbitrary rank.

Visual [Beginner]

An irreducible variety $X$ with a degeneracy locus $D$ shown as a connected shaded region. The map between ample bundles drops rank precisely on $D$ . The connectedness of $D$ is the content of the theorem.

The ample condition on the bundles is the positivity hypothesis that forces the degeneracy locus to be connected rather than scattered across disconnected components.

Worked example [Beginner]

On $P^{n}$ , consider two copies of the tautological quotient bundle $Q$ (which is ample). A section $s : O \to Q^{\oplus 2}$ gives two sections $s_{1}, s_{2}$ of $Q$ . The degeneracy locus where $s_{1} = s_{2} = 0$ is the intersection of the zero loci of two ample sections.

For $n = 2$ : $Q$ has rank 1 on $P^{2}$ (it is $O (1)$ ). The zero locus of a section of $O (1)$ is a line. Two general lines in $P^{2}$ meet at one point (connected). The Fulton-Lazarsfeld theorem guarantees this connectedness for any two ample sections.

Check your understanding [Beginner]

Exercise (easy, multiple choice).

The Fulton-Lazarsfeld connectedness theorem states that degeneracy loci of maps between ample vector bundles are:

A. Always smooth

B. Always irreducible

C. Connected when the expected codimension equals the dimension of the base

D. Always empty

Hint

The theorem gives a topological connectedness result, not smoothness or irreducibility.

Answer

C. Feedback-correct: the degeneracy locus is connected when the expected codimension (from the Porteous formula) equals the dimension of the base, so the degeneracy locus is expected to be 0-dimensional (finitely many points, which must form a connected set, hence a single point). More generally, when the expected dimension is $\geq 0$ , the degeneracy locus is connected. Feedback-wrong: A is false (degeneracy loci are generally singular); B is too strong (connected does not imply irreducible); D is false.

Formal definition [Intermediate+]

Let $X$ be an irreducible projective variety of dimension $n$ , and let $E$ and $F$ be vector bundles on $X$ of ranks $e$ and $f$ .

Definition (ample vector bundle). A vector bundle $E$ on $X$ is ample if the tautological line bundle $O_{P (E)} (1)$ on the projectivisation $P (E)$ is ample. Equivalently, for any coherent sheaf $F$ on $X$ , $H^{i} (X, Sym^{m} (E) \otimes F) = 0$ for $m ≫ 0$ and $i > 0$ .

Theorem (Fulton-Lazarsfeld connectedness). Let $ϕ : E \to F$ be a morphism of vector bundles on an irreducible projective variety $X$ , with $E$ globally generated and $F^{\lor}$ ample (i.e., $F$ is "anti-ample" in the dual sense, or more generally, $E - F$ is ample in the sense of $Q$ -twists). If the degeneracy locus $$ D_k(\phi) = {x \in X : \mathrm{rank}(\phi_x) \leq k} $$ has the expected codimension $(e - k) (f - k)$ , then $D_{k} (ϕ)$ is connected when $dim D_{k} \geq 1$ , and non-empty when $dim D_{k} = 0$ .

Theorem (Fulton-Lazarsfeld positivity). If $E$ is an ample vector bundle of rank $e$ on a projective variety $X$ , and $P$ is a positive homogeneous polynomial of degree $d \leq dim X$ in the Chern classes of $E$ (i.e., $P$ is a non-negative linear combination of Schur polynomials), then $$ \int_X P(c(E)) > 0. $$

Counterexamples to common slips.

Ampleness of $E$ alone is not sufficient. The Fulton-Lazarsfeld connectedness theorem requires a positivity hypothesis on the map $ϕ : E \to F$ , typically that $E - F$ is ample or that $E$ is globally generated and $F^{\lor}$ is ample.
Connectedness is not irreducibility. The degeneracy locus can be connected but reducible (e.g., a union of two curves meeting at a point).
Expected codimension is necessary. If the degeneracy locus has excess dimension, the connectedness conclusion can fail.

Key theorem with proof [Intermediate+]

Theorem (Fulton-Lazarsfeld). Let $E$ and $F$ be vector bundles of ranks $e$ and $f$ on an irreducible projective variety $X$ of dimension $n$ , with $E$ globally generated and $F$ ample. Let $ϕ : E \to F$ be a bundle map, and let $D_{k} (ϕ)$ be the $k$ -th degeneracy locus with expected codimension $(e - k) (f - k)$ . Then $D_{k} (ϕ)$ is connected (if $dim D_{k} \geq 1$ ) or a single point (if $dim D_{k} = 0$ ).

Proof (sketch).

Step 1: Reduction to the zero-locus case. On the projectivisation $P (E)$ , the map $ϕ$ induces a section $\tilde{ϕ}$ of the bundle $S^{\lor} \otimes π^{*} F$ where $S$ is the tautological subbundle and $π : P (E) \to X$ is the projection. The degeneracy locus $D_{k} (ϕ)$ pulls back to the zero locus of $\tilde{ϕ}$ (up to a blow-down).

Step 2: Ample bundles on projective space bundles. Since $F$ is ample on $X$ , the bundle $π^{*} F$ is ample on $P (E)$ . The tensor product $S^{\lor} \otimes π^{*} F$ is ample (tensor product of a globally generated line bundle with an ample bundle is ample).

Step 3: Barth-type connectedness. A section of an ample vector bundle on an irreducible projective variety either vanishes identically or has connected zero locus, by the Fulton-Hansen connectedness theorem applied to the embedding defined by the ample bundle.

Step 4: Degeneracy locus structure. The zero locus of $\tilde{ϕ}$ maps onto $D_{k} (ϕ)$ with fibres that are connected (they are linear subspaces of $P (E_{x})$ ). The image of a connected set under a continuous map is connected. Hence $D_{k} (ϕ)$ is connected. $□$

Bridge. The Fulton-Lazarsfeld theorem connects the intersection-theoretic Porteous formula 04.04.13 with topological connectedness, using the ampleness hypothesis to turn the Porteous class (a Schur polynomial in Chern classes) into a positive number, guaranteeing non-emptiness. The same positivity principle controls the Brill-Noether loci $W_{d}^{r}$ 04.04.08: when the Brill-Noether number $ρ \geq 1$ , the locus $W_{d}^{r}$ is connected, a result proved by applying Fulton-Lazarsfeld to the evaluation map on the Poincaré bundle (using the ampleness of the theta divisor on the Jacobian). This connectedness feeds into Martens' theorem 04.04.10 and the gonality classification 04.04.11, where the connectedness of $W_{d}^{1}$ for $ρ \geq 1$ guarantees the existence of the gonality map on general curves.

Exercises [Intermediate+]

Exercise 2 (medium, symbolic).

Apply the Fulton-Lazarsfeld theorem to the Brill-Noether locus $W_{d}^{r}$ on a general curve of genus $g$ , showing it is connected when $ρ \geq 1$ .

Hint

$W_{d}^{r}$ is the degeneracy locus of the evaluation map on the Poincaré bundle. Identify the ampleness hypothesis using the theta divisor.

Answer

The Brill-Noether locus $W_{d}^{r}$ is the degeneracy locus of the map $ev : H^{0} (L) \otimes O \to L$ on the Jacobian $Pic^{d} (C)$ , using the Poincaré bundle. The theta divisor $Θ \subset Pic^{g - 1} (C)$ is ample (it defines a principal polarisation), and the Poincaré bundle restricted to ${L} \times C$ is $L$ itself. The ampleness of $Θ$ gives the positivity hypothesis needed for Fulton-Lazarsfeld.

When $ρ = g - (r + 1) (g - d + r) \geq 1$ , the expected dimension of $W_{d}^{r}$ is $ρ \geq 1$ . By the Porteous formula 04.04.13, the class of $W_{d}^{r}$ is a Schur polynomial in the Chern classes of the Poincaré bundle, which is positive by the Fulton-Lazarsfeld positivity theorem. Hence $W_{d}^{r}$ is non-empty and connected.

Exercise 3 (hard, symbolic).

Prove that if $E$ is an ample vector bundle on a projective variety $X$ of dimension $n$ , and $s$ is a global section of $E$ , then the zero locus $Z (s)$ is connected when $dim Z (s) \geq 1$ .

Hint

Use the Fulton-Hansen connectedness theorem: a map from an irreducible projective variety to $P^{N}$ whose image has dimension $\geq 2$ has connected inverse image of any linear subspace.

Answer

Since $E$ is ample, the bundle $O_{P (E)} (1)$ is ample on $P (E)$ . A section $s$ of $E$ corresponds to a section $\tilde{s}$ of $O_{P (E)} (1)$ , whose zero locus is the projectivisation of the kernel of $s$ , denoted $P (ker s) \subset P (E)$ .

The zero locus $Z (s) \subset X$ is the image of $P (ker s)$ under the projection $π : P (E) \to X$ . Since $P (E)$ is irreducible (being a projective bundle over an irreducible base) and $\tilde{s}$ is a section of an ample line bundle, the complement $P (E) ∖ Z (\tilde{s})$ is an affine variety, hence $P (ker s)$ is an ample divisor on $P (E)$ .

By the Barth-type theorem (the complement of an ample divisor in an irreducible projective variety is connected in codimension 1), or more directly by the Lefschetz hyperplane theorem in the topological setting, $P (ker s)$ is connected. The projection $π$ maps the connected set $P (ker s)$ onto $Z (s)$ , so $Z (s)$ is connected.

Advanced results [Master]

Fulton-Hansen connectedness theorem. The foundational result underlying Fulton-Lazarsfeld: if $f : X \to P^{N}$ is a morphism from an irreducible projective variety with $dim f (X) \geq 2$ , and $L \subset P^{N}$ is a linear subspace, then $f^{- 1} (L)$ is connected. This is a consequence of the Fulton-Hansen theorem (1979) and generalises the Barth-Larsen theorem.

Positive Schur polynomials. The Fulton-Lazarsfeld positivity theorem: if $E$ is an ample vector bundle and $Δ_{λ}$ is a Schur polynomial corresponding to a partition $λ$ of weight $\leq dim X$ , then $\int_{X} Δ_{λ} (c (E)) > 0$ . This is the algebraic engine behind the connectedness theorem: the Porteous class is a positive Schur polynomial, hence the degeneracy locus has positive degree.

Applications to moduli. The connectedness of Brill-Noether loci $W_{d}^{r}$ for $ρ \geq 1$ is a direct application. More refined applications include the connectedness of the Petri locus (the set of curves violating the Petri condition) inside $M_{g}$ , and the connectedness of higher-rank Brill-Noether loci in the moduli of vector bundles on curves.

Lazarsfeld's ampleness criterion. Lazarsfeld used the Fulton-Lazarsfeld machinery to prove that the vector bundle $E$ on a smooth curve $C$ contained in a K3 surface is "as positive as possible" — the restriction theorem, which is the key input for his proof of the Brill-Noether-Petri theorem 04.04.16.

Synthesis. The Fulton-Lazarsfeld connectedness theorem is the positivity backbone of the Brill-Noether theory of curves, unifying four previously disconnected results. The first is the Porteous formula 04.04.13 which computes the fundamental class of the degeneracy locus as a Schur polynomial in Chern classes. The second is the Fulton-Hansen connectedness for maps to projective space, which is the geometric engine proving that ample-section zero loci are connected. The third is the positivity of Schur polynomials for ample bundles, which guarantees that the Porteous class is non-zero (hence the degeneracy locus is non-empty). The fourth is the application to Brill-Noether theory 04.04.08, where the connectedness of $W_{d}^{r}$ for $ρ \geq 1$ follows from applying Fulton-Lazarsfeld to the evaluation map on the Poincaré bundle. The same positivity framework underpins Lazarsfeld's restriction theorem on K3 surfaces 04.04.16 and the dimension bounds of Martens' theorem 04.04.10.

Full proof set [Master]

The proof of the Fulton-Lazarsfeld connectedness theorem is sketched in the Key theorem section. The full proof proceeds via the following steps: (1) pull back the bundle map to the projectivisation $P (E)$ , converting the degeneracy locus to a zero locus of a section of an ample vector bundle; (2) apply the Fulton-Hansen connectedness theorem to the embedding of $P (E)$ by the ample line bundle $O_{P (E)} (1)$ ; (3) deduce that the zero locus of the section is connected; and (4) project back to $X$ , preserving connectedness since the fibres of the projection are connected (they are projective spaces).

The Fulton-Hansen theorem itself is proved via the Lefschetz hyperplane theorem in the analytic setting, or via the Barth-Larsen theorem in the algebraic setting: a finite morphism $f : X \to P^{N}$ from an irreducible variety has the property that $f^{- 1} (L)$ is connected for any linear subspace $L$ of codimension $\leq N - 2$ .

Connections [Master]

Determinantal varieties and Porteous 04.04.13 — the Porteous formula computes the class of the degeneracy locus; Fulton-Lazarsfeld proves this class is positive for ample bundles, hence the locus is non-empty and connected.
Petri map and Gieseker-Petri 04.04.08 — the connectedness of $W_{d}^{r}$ for $ρ \geq 1$ is a direct application of Fulton-Lazarsfeld to the Brill-Noether locus, via the Petri map tangent-space identification.
Martens' theorem 04.04.10 — Martens' dimension bound $dim W_{d}^{r} \leq d - 2 r - 2$ combined with Fulton-Lazarsfeld connectedness gives the complete topological picture of Brill-Noether loci.
Gonality 04.04.11 — the non-emptiness of $W_{d}^{1}$ for $ρ \geq 0$ (a consequence of Fulton-Lazarsfeld positivity) guarantees the existence of the gonality map on general curves.
Lazarsfeld K3 proof 04.04.16 — the ampleness of vector bundles on K3 surfaces and the Fulton-Lazarsfeld positivity framework are the technical inputs for Lazarsfeld's proof of the Petri theorem.

Historical & philosophical context [Master]

William Fulton and Robert Lazarsfeld proved the connectedness theorem in the 1981 paper Positive polynomials for ample vector bundles (Ann. Math. 118, 35-60) ^{[Fulton-Lazarsfeld 1981]}. The paper introduced the general principle that Schur polynomials in the Chern classes of ample vector bundles are numerically positive, and deduced connectedness of degeneracy loci as a consequence.

The work built on two foundational results: the Fulton-Hansen connectedness theorem (1979), which proved that finite maps from irreducible varieties to projective space have connected inverse images of linear subspaces; and the Kleiman criterion (1974) for ampleness, which characterises ample bundles in terms of the positivity of their Chern numbers.

The philosophical content: positivity in algebraic geometry (ampleness, nefness, bigness) has topological consequences. An ample vector bundle is "curved enough" that its degeneracy loci cannot break apart into disconnected pieces. The Fulton-Lazarsfeld theorem is the quantitative expression of this principle for determinantal conditions.

Bibliography [Master]

Fulton, W.; Lazarsfeld, R. (1981). Positive polynomials for ample vector bundles. Annals of Mathematics 118, 35-60. [Originator paper.]
Fulton, W.; Hansen, J. (1979). A connectedness theorem for projective varieties, with applications to intersections and singularities of mappings. Ann. Math. 110, 159-166. [Fulton-Hansen connectedness.]
Fulton, W. Intersection Theory. Springer, Ch. 12. [Standard treatment of connectedness.]
Lazarsfeld, R. Positivity in Algebraic Geometry I. Springer, Ch. 7 and 8. [Degeneracy loci and positivity.]

Prerequisites

04.04.13
04.04.08

Tier anchors

beginner: Fulton-Lazarsfeld 1981 informal — degeneracy loci of ample bundles are connected
intermediate: Fulton-Lazarsfeld 1981; Fulton — Intersection Theory Ch. 12; Lazarsfeld — Positivity in Algebraic Geometry
master: Fulton-Lazarsfeld 1981 *Positive polynomials for ample vector bundles* (Ann. Math. 118, originator); Fulton — Intersection Theory Ch. 12; Lazarsfeld — Positivity in Algebraic Geometry I

References

TODO_REF
Fulton-Lazarsfeld 1981 — Positive polynomials for ample vector bundles · Ann. Math. 118, 35-60; originator
TODO_REF
Fulton — Intersection Theory · Springer, Ch. 12 connectedness
TODO_REF
Lazarsfeld — Positivity in Algebraic Geometry I · Springer, Ch. 7 and 8 degeneracy loci and positivity

Reviewer

TBD

Estimated time

beginner: 14m
intermediate: 40m
master: 75m