Dimension Constructible Images and Dimensions of Fibres: Examples

ExampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

The coordinate cross has two one-dimensional components

Example

The coordinate cross $C = V (x y) \subseteq A_{k}^{2}$ has two irreducible components, both affine lines. Its global dimension is one, and $\dim_{p} C = 1$ at every closed point, including the origin.

Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

If a Noetherian space $T$ is a finite union of closed subsets $T_{1}, \dots, T_{m}$ , then $\dim T = \max_{i} \dim T_{i}$ . For $m = 0$ both sides are $- \infty$ . (Dimension of a finite closed union).

[F2]

For a classical variety $X$ , let $\dim X$ be its chain dimension. If $X_{1}, \dots, X_{m}$ are its irreducible components and $x \in X$ is a closed point, define $\dim_{x} X = \max_{x \in X_{i}} \dim X_{i}$ . The indexing family is nonempty. Say that $X$ has pure dimension $d$ if every irreducible component has dimension $d$ ; the condition on components is vacuous for the empty variety, whose dimension is nevertheless $- \infty$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Global and local dimension of classical varieties).

[F3]

For every integer $n \geq 0$ , $\dim A_{k}^{n} = \dim P_{k}^{n} = n$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Affine and projective n-space have dimension n).

Verification

1.1

The equation $x y = 0$ means $x = 0$ or $y = 0$ because $k$ is a field. Thus $C$ is the union of the two coordinate lines. Each is an affine line, irreducible of dimension one, and neither contains the other, so these are exactly the components.

F3

2.1

The finite-union formula gives global dimension one. At a point other than the origin exactly one component passes through the point; at the origin both do. The maximum of their dimensions is one in either case, which is the stated local dimension.

F1F2step 1.1∎

ExampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Elementary fibres: empty, points, and affine lines

Example

For $H = V (x y - 1) \subseteq A_{k}^{2}$ , projection $H \to A_{k}^{1}$ onto $x$ has fibre ${(a, a^{- 1})}$ when $a \neq 0$ and empty fibre at $a = 0$ . Their dimensions are zero and $- \infty$ respectively. By contrast, every fibre of $A^{2} \to A^{1}$ , $(x, y) \mapsto x$ , is an affine line of dimension one.

Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

For a morphism $f : X \to Y$ of classical varieties and a closed point $y \in Y$ , let $X_{y} = f^{- 1} (y)$ have its reduced closed-subvariety structure. Its dimension is the chain dimension, with $\dim X_{y} = - \infty$ if the fibre is empty. On affine charts $V \subseteq Y$ containing $y$ and $U \subseteq f^{- 1} (V)$ , writing $A = k [V]$ and $B = k [U]$ , the fibre chart has coordinate ring $B / \sqrt{m_{y} B}$ . Here general morphisms have the locally ringed-space meaning; the earlier affine morphism definition applies to the restrictions $U \to V$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Reduced closed-point fibres and their dimension).

[F2]

For every integer $n \geq 0$ , $\dim A_{k}^{n} = \dim P_{k}^{n} = n$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Affine and projective n-space have dimension n).

Verification

1.1

Fixing $x = a$ in $x y = 1$ gives $a y = 1$ . If $a \neq 0$ the unique solution is $y = a^{- 1}$ ; if $a = 0$ it has no solution. These reduced fibres are respectively a point and the empty variety, of dimensions zero and $- \infty$ .

F1F2

2.1

For the full-plane projection fixing $x = a$ leaves $y$ arbitrary. The regular maps $y \mapsto (a, y)$ and $(a, y) \mapsto y$ are inverse isomorphisms of the fibre with $A^{1}$ . Thus its dimension is one for every $a$ , including zero.

F2step 1.1∎

ExampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

The family xy=t has constant dimension and a reducible special fibre

Example

For the morphism $V (x y - t) \subseteq A_{k}^{3} \to A_{k}^{1}$ , $(x, y, t) \mapsto t$ , the fibre at $a \neq 0$ is $G_{m}$ , whereas the fibre at zero is the coordinate cross. Every fibre is nonempty of pure dimension one.

Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F2]

For every integer $n \geq 0$ , $\dim A_{k}^{n} = \dim P_{k}^{n} = n$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Affine and projective n-space have dimension n).

[F3]

If $U$ is a nonempty open of an irreducible classical variety $X$ , then $\dim U = \dim X$ . Every proper closed subvariety $Z ⊊ X$ has $\dim Z < \dim X$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Nonempty opens preserve irreducible dimension).

[F4]

If a Noetherian space $T$ is a finite union of closed subsets $T_{1}, \dots, T_{m}$ , then $\dim T = \max_{i} \dim T_{i}$ . For $m = 0$ both sides are $- \infty$ . (Dimension of a finite closed union).

[F5]

For a morphism $f : X \to Y$ of classical varieties and a closed point $y \in Y$ , let $X_{y} = f^{- 1} (y)$ have its reduced closed-subvariety structure. Its dimension is the chain dimension, with $\dim X_{y} = - \infty$ if the fibre is empty. On affine charts $V \subseteq Y$ containing $y$ and $U \subseteq f^{- 1} (V)$ , writing $A = k [V]$ and $B = k [U]$ , the fibre chart has coordinate ring $B / \sqrt{m_{y} B}$ . Here general morphisms have the locally ringed-space meaning; the earlier affine morphism definition applies to the restrictions $U \to V$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Reduced closed-point fibres and their dimension).

Verification

1.1

The total space is isomorphic to $A^{2}$ by $(x, y) \mapsto (x, y, x y)$ , with inverse forgetting $t$ . Its fibre at $a$ is the reduced zero locus $x y = a$ . For $a \neq 0$ , $x \mapsto (x, a / x, a)$ identifies it with $G_{m}$ , a nonempty open of $A^{1}$ of dimension one and irreducible.

F2F3F5

2.1

At $a = 0$ , $x y = 0$ is the union of the two coordinate axes, each an affine line; these are its two irreducible components. The finite-union formula gives dimension one, and each component has that dimension. Thus the special fibre is reducible but still pure of dimension one. All parameters have a nonempty fibre.

F2F4step 1.1∎

ExampleConstruction: Literature-sourcedVerification: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

The map (x,y) to (x,xy) has a jumping fibre

Example

The morphism $f : A_{k}^{2} \to A_{k}^{2}$ , $(x, y) \mapsto (u, v) = (x, x y)$ , has image $D (u) \cup {(0, 0)}$ . Its fibres are one point when $u \neq 0$ , an affine line over $(0, 0)$ , and empty over $(0, v)$ with $v \neq 0$ . The image is constructible and is not locally closed.

Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

For a morphism $f : X \to Y$ of classical varieties and a closed point $y \in Y$ , let $X_{y} = f^{- 1} (y)$ have its reduced closed-subvariety structure. Its dimension is the chain dimension, with $\dim X_{y} = - \infty$ if the fibre is empty. On affine charts $V \subseteq Y$ containing $y$ and $U \subseteq f^{- 1} (V)$ , writing $A = k [V]$ and $B = k [U]$ , the fibre chart has coordinate ring $B / \sqrt{m_{y} B}$ . Here general morphisms have the locally ringed-space meaning; the earlier affine morphism definition applies to the restrictions $U \to V$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Reduced closed-point fibres and their dimension).

[F2]

A subset $S$ of a classical variety $X$ is locally closed if $S = U \cap Z$ for some open $U \subseteq X$ and closed $Z \subseteq X$ . A subset is constructible if it is a finite union of locally closed subsets. The empty union is allowed, so $\emptyset$ is constructible. Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Locally closed and constructible subsets).

[F3]

For every integer $n \geq 0$ , $\dim A_{k}^{n} = \dim P_{k}^{n} = n$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Affine and projective n-space have dimension n).

Verification

1.1

The fibre equations are $x = u$ and $u y = v$ . For $u \neq 0$ there is exactly one solution $(u, v / u)$ , of dimension zero. If $u = v = 0$ , $y$ is free, so the fibre is an affine line of dimension one. If $u = 0$ and $v \neq 0$ there is no solution. This proves the image description and the empty-fibre dimension $- \infty$ .

F1F3

2.1

The image is a union of an open subset and a closed point, hence constructible. It is dense since $D (u)$ is dense in the irreducible affine plane. If this image were locally closed, writing it as open intersect closed and taking closure would show it open in the plane. But any open neighborhood of the origin meets the line $u = 0$ in a nonempty open subset of that line. A proper closed subset of an affine line is finite by the polynomial root bound, so this neighborhood contains a point $(0, v)$ with $v \neq 0$ , outside the image. Therefore the image is not open and not locally closed.

F2step 1.1∎

CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

A morphism image need not be closed

Statement refuted

False claim: every regular morphism of classical varieties has closed image.

Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points.

Facts & Assumptions

Given: The objects and hypotheses in the statement refuted, for the explicit witness below.

[F1]

A subset $S$ of a classical variety $X$ is locally closed if $S = U \cap Z$ for some open $U \subseteq X$ and closed $Z \subseteq X$ . A subset is constructible if it is a finite union of locally closed subsets. The empty union is allowed, so $\emptyset$ is constructible. Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Locally closed and constructible subsets).

[F2]

For a morphism $f : X \to Y$ of classical varieties and a closed point $y \in Y$ , let $X_{y} = f^{- 1} (y)$ have its reduced closed-subvariety structure. Its dimension is the chain dimension, with $\dim X_{y} = - \infty$ if the fibre is empty. On affine charts $V \subseteq Y$ containing $y$ and $U \subseteq f^{- 1} (V)$ , writing $A = k [V]$ and $B = k [U]$ , the fibre chart has coordinate ring $B / \sqrt{m_{y} B}$ . Here general morphisms have the locally ringed-space meaning; the earlier affine morphism definition applies to the restrictions $U \to V$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Reduced closed-point fibres and their dimension).

Counterexample

1.1

Take projection $p : H = V (x y - 1) \to A^{1}$ onto $x$ . Its equations have a solution precisely when $x \neq 0$ : the solution is $y = x^{- 1}$ . Thus $p (H) = D (x)$ , a constructible open subset.

F1F2

2.1

The set $D (x)$ is proper since it omits zero, and is dense: any one-variable polynomial vanishing at all its infinitely many points is the zero polynomial. Therefore its image is not closed, refuting the claim.

step 1.1∎

CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Finite fibres do not imply a module-finite map

Statement refuted

False claim: a morphism of affine classical varieties with finite fibres must be module-finite.

Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points.

Facts & Assumptions

Given: The objects and hypotheses in the statement refuted, for the explicit witness below.

[F1]

A morphism $f : X \to Y$ of classical varieties is quasi-finite if every closed-point fibre $X_{y}$ is a finite set; empty fibres are allowed. Classical morphisms here are of finite type: for an affine target chart and an affine source chart above it, any finite set of $k$ -algebra generators of the source ring also generates it over the target ring. The inverse image has a finite affine cover because it is an open of a Noetherian variety. Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Quasi-finite classical morphisms).

[F2]

For affine classical algebraic sets $X, Y$ , call a morphism $f : X \to Y$ module-finite if $k [X]$ , via pullback, is a finitely generated $k [Y]$ -module. This is the affine module criterion. Empty affine sets are allowed, with zero coordinate ring; the definition does not assert a global affine-preimage criterion for arbitrary varieties. Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Module-finite affine maps for the quasi-finite comparison).

[F3]

For a morphism $f : X \to Y$ of classical varieties and a closed point $y \in Y$ , let $X_{y} = f^{- 1} (y)$ have its reduced closed-subvariety structure. Its dimension is the chain dimension, with $\dim X_{y} = - \infty$ if the fibre is empty. On affine charts $V \subseteq Y$ containing $y$ and $U \subseteq f^{- 1} (V)$ , writing $A = k [V]$ and $B = k [U]$ , the fibre chart has coordinate ring $B / \sqrt{m_{y} B}$ . Here general morphisms have the locally ringed-space meaning; the earlier affine morphism definition applies to the restrictions $U \to V$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Reduced closed-point fibres and their dimension).

Counterexample

1.1

Take $j : G_{m} ↪ A^{1}$ , with ring map $k [t] ↪ k [t, t^{- 1}]$ . Its fibres are a singleton at each nonzero parameter and empty at zero, so it is quasi-finite. The inverse map between $G_{m}$ and $V (t s - 1)$ identifies its coordinate ring with the indicated Laurent ring.

F1F3

2.1

If finitely many Laurent polynomials generated $k [t, t^{- 1}]$ as a $k [t]$ -module, there would be $M \geq 0$ such that every exponent in all these generators was at least $- M$ . Multiplication by polynomials and finite addition cannot introduce a smaller exponent. Thus their span cannot contain $t^{- (M + 1)}$ , a contradiction. The module-finiteness criterion fails.

F2step 1.1∎

ExampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Plane curves meet; common components change the dimension

Example

Two distinct lines in $P_{k}^{2}$ intersect in one point. The reducible curves $V_{+} (X Y)$ and $V_{+} (X Z)$ intersect in the line $X = 0$ together with the point $[1 : 0 : 0]$ . In contrast, two distinct irreducible projective plane curves have a nonempty finite intersection.

Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

Let $X, Y \subseteq P_{k}^{n}$ be irreducible closed subvarieties. Every nonempty irreducible component $Z$ of $X \cap Y$ satisfies $\dim Z \geq \dim X + \dim Y - n$ . If $\dim X + \dim Y \geq n$ , then $X \cap Y \neq \emptyset$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Projective intersection dimension and nonemptiness).

[F2]

If a Noetherian space $T$ is a finite union of closed subsets $T_{1}, \dots, T_{m}$ , then $\dim T = \max_{i} \dim T_{i}$ . For $m = 0$ both sides are $- \infty$ . (Dimension of a finite closed union).

[F3]

For every integer $n \geq 0$ , $\dim A_{k}^{n} = \dim P_{k}^{n} = n$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Affine and projective n-space have dimension n).

[F4]

If $U$ is a nonempty open of an irreducible classical variety $X$ , then $\dim U = \dim X$ . Every proper closed subvariety $Z ⊊ X$ has $\dim Z < \dim X$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Nonempty opens preserve irreducible dimension).

Verification

1.1

Two distinct lines are defined by independent linear forms on $k^{3}$ . Their common kernel has vector dimension one, so its projectivization is a single point. For the displayed reducible curves, the equations $X Y = X Z = 0$ imply either $X = 0$ , giving the whole line, or $X \neq 0$ and $Y = Z = 0$ , giving $[1 : 0 : 0]$ . This point is outside the line. The intersection has dimension one, as a finite closed union of a line and a point.

F2F3

2.1

For distinct irreducible plane curves $C, D$ of dimension one, the projective intersection theorem ensures nonemptiness because $1 + 1 = 2$ . Their intersection is a proper closed subset of $C$ : otherwise $C \subseteq D$ , and a proper closed subset of irreducible $D$ could not have dimension one. Thus all components of $C \cap D$ have dimension zero by proper-closed dimension drop. There are finitely many components, each a point (a larger irreducible closed set would contain a singleton chain of length one). Consequently the intersection is finite.

F1F4step 1.1∎

CounterexampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Dimension zero for the empty set loses the empty-fibre distinction

Statement refuted

Incompatibility to refute: adopt $\dim \emptyset = 0$ while retaining the assertion $\dim T \geq 0$ if and only if $T \neq \emptyset$ . The convention $\dim \emptyset = - \infty$ makes that assertion, and the empty maximum, literal; no uniqueness among all possible dimension conventions is claimed.

Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points.

Facts & Assumptions

Given: The objects and hypotheses in the statement refuted, for the explicit witness below.

[F1]

For a Noetherian topological space $T$ , define $\dim T$ as the supremum of the lengths $s$ of strict chains $Z_{0} ⊊ \dots ⊊ Z_{s}$ of nonempty irreducible closed subsets of $T$ . Thus a one-member chain has length zero. Set $\dim \emptyset = - \infty$ , and allow $\dim T = + \infty$ . The supremum of an empty family of dimensions is $- \infty$ . (Chain dimension and the empty-space convention).

[F2]

For a morphism $f : X \to Y$ of classical varieties and a closed point $y \in Y$ , let $X_{y} = f^{- 1} (y)$ have its reduced closed-subvariety structure. Its dimension is the chain dimension, with $\dim X_{y} = - \infty$ if the fibre is empty. On affine charts $V \subseteq Y$ containing $y$ and $U \subseteq f^{- 1} (V)$ , writing $A = k [V]$ and $B = k [U]$ , the fibre chart has coordinate ring $B / \sqrt{m_{y} B}$ . Here general morphisms have the locally ringed-space meaning; the earlier affine morphism definition applies to the restrictions $U \to V$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Reduced closed-point fibres and their dimension).

Counterexample

1.1

The fibre at zero of $V (x y - 1) \to A^{1}$ , $(x, y) \mapsto x$ , is empty since $0 \cdot y = 1$ has no solution. Giving it dimension zero makes $\dim T \geq 0$ true while $T \neq \emptyset$ is false. Thus the two proposed rules are incompatible.

F2

2.1

With the chain convention, the empty space has no chain and dimension $- \infty$ . If a Noetherian space $T$ is nonempty, choose a point; its closure is a nonempty irreducible closed subset and provides a length-zero chain, so $\dim T \geq 0$ . Conversely $\dim T \geq 0$ excludes the empty space. Defining the supremum and maximum over the empty family as $- \infty$ also makes the finite-closed-union formula consistent for an empty cover.

F1step 1.1∎

CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

The ambient affine-space hypothesis matters

Statement refuted

False claim: for irreducible closed subsets $P, Q$ of any irreducible classical ambient variety $W$ , every nonempty component of $P \cap Q$ has dimension at least $\dim P + \dim Q - \dim W$ .

Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points.

Facts & Assumptions

Given: The objects and hypotheses in the statement refuted, for the explicit witness below.

[F1]

Let $X$ be irreducible affine and $0 \neq f \in k [X]$ be a nonunit. Then $V_{X} (f)$ is nonempty and every irreducible component has dimension $\dim X - 1$ , hence codimension one. Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (A nontrivial principal section has pure codimension one).

[F2]

For every integer $n \geq 0$ , $\dim A_{k}^{n} = \dim P_{k}^{n} = n$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Affine and projective n-space have dimension n).

[F3]

For irreducible closed $X, Y \subseteq A_{k}^{n}$ , every nonempty irreducible component $Z$ of $X \cap Y$ satisfies $\dim Z \geq \dim X + \dim Y - n$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Affine intersection bound via the diagonal).

Counterexample

1.1

Let $W = V (x_{1} x_{4} - x_{2} x_{3}) \subseteq A^{4}$ . The polynomial is irreducible: viewing it as a primitive polynomial in $x_{4}$ over $k [x_{1}, x_{2}, x_{3}]$ , its coefficients $x_{1}$ and $- x_{2} x_{3}$ have no nonunit common factor, and over the fraction field it is linear. Gauss reduction proves irreducibility in the polynomial ring; equivalently a factor independent of $x_{4}$ would divide both coefficients and be a unit. Thus $W$ is irreducible, and the principal theorem in affine four-space gives $\dim W = 3$ .

F1F2

2.1

The subspaces $P = V (x_{2}, x_{4})$ and $Q = V (x_{1}, x_{3})$ lie in $W$ and are affine planes, each of dimension two. Their intersection is exactly the origin, of dimension zero. The claimed ambient bound would require $0 \geq 2 + 2 - 3 = 1$ , which is false. The valid affine-space bound instead uses ambient $A^{4}$ and gives $0 \geq 2 + 2 - 4 = 0$ .

F2F3step 1.1∎

ExampleConstruction: Literature-sourcedVerification: AI-adaptedaudited 2026-09-07Open item page →

Fibre dimensions of a family of homogeneous linear systems

Example

Let $A (y)$ be an $m \times n$ matrix of regular functions on a classical variety $Y$ , with $m, n \geq 0$ , and $W = {(y, x) \in Y \times A^{n} : A (y) x = 0}$ . Then $\dim W_{y} = n - rank A (y)$ . For every integer $r$ , the locus ${y : \dim W_{y} \geq r}$ is closed: it is all of $Y$ for $r \leq 0$ , empty for $r > n$ , and otherwise is cut out by the minors of size $n - r + 1$ , with an absent family of minors imposing no conditions. No irreducibility or global dimension formula for $W$ is asserted.

Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

For a morphism $f : X \to Y$ of classical varieties and a closed point $y \in Y$ , let $X_{y} = f^{- 1} (y)$ have its reduced closed-subvariety structure. Its dimension is the chain dimension, with $\dim X_{y} = - \infty$ if the fibre is empty. On affine charts $V \subseteq Y$ containing $y$ and $U \subseteq f^{- 1} (V)$ , writing $A = k [V]$ and $B = k [U]$ , the fibre chart has coordinate ring $B / \sqrt{m_{y} B}$ . Here general morphisms have the locally ringed-space meaning; the earlier affine morphism definition applies to the restrictions $U \to V$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Reduced closed-point fibres and their dimension).

[F2]

For every integer $n \geq 0$ , $\dim A_{k}^{n} = \dim P_{k}^{n} = n$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Affine and projective n-space have dimension n).

[F3]

Products of nonempty classical varieties exist in the category of classical varieties, and $\dim (X \times_{k} Y) = \dim X + \dim Y$ . If both factors are irreducible, their product is irreducible. Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Dimensions add under products).

Verification

1.1

The product exists, and on its affine base charts the entries of $A (y) x$ are regular polynomial expressions. Their common zeros give a reduced closed subvariety $W$ . Fix $y$ . Elementary row and column changes over $k$ reduce $A (y)$ to a block matrix with an identity block of size $q = rank A (y)$ and zeros elsewhere. The inverse linear coordinate changes identify its kernel, as an affine algebraic set, with $A^{n - q}$ . Thus the fibre has dimension $n - q$ and always contains zero.

F1F2F3

2.1

For $1 \leq s \leq \min (m, n)$ , rank is at least $s$ exactly when some $s \times s$ minor is nonzero. One implication follows from the independence of its columns. For the other, choose $s$ independent columns; the resulting injective map $k^{s} \to k^{m}$ has $s$ independent coordinate row functionals, giving such a minor. Therefore rank at most $s - 1$ is equivalent to vanishing of all size- $s$ minors. For $1 \leq r \leq n$ , take $s = n - r + 1$ : these minors are regular functions, hence define a closed locus. If $s > m$ all such minors are absent and the rank bound holds automatically.

step 1.1

3.1

Every fibre is nonempty and has dimension between zero and $n$ , so $r \leq 0$ gives all of $Y$ and $r > n$ gives the empty locus. If $n = 0$ , every fibre is a point; if $m = 0$ , there are no equations and every fibre is $A^{n}$ . The zero matrix gives that latter fibre too. These verify all boundary conventions without assuming the total space is irreducible.

F2step 1.1step 2.1∎

Dimension Constructible Images and Dimensions of Fibres — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

The coordinate cross has two one-dimensional components

Example

Facts & Assumptions

Verification

Elementary fibres: empty, points, and affine lines

Example

Facts & Assumptions

Verification

The family xy=t has constant dimension and a reducible special fibre

Example

Facts & Assumptions

Verification

The map (x,y) to (x,xy) has a jumping fibre

Example

Facts & Assumptions

Verification

A morphism image need not be closed

Statement refuted

Facts & Assumptions

Counterexample

Finite fibres do not imply a module-finite map

Statement refuted

Facts & Assumptions

Counterexample

Plane curves meet; common components change the dimension

Example

Facts & Assumptions

Verification

Dimension zero for the empty set loses the empty-fibre distinction

Statement refuted

Facts & Assumptions

Counterexample

The ambient affine-space hypothesis matters

Statement refuted

Facts & Assumptions

Counterexample

Fibre dimensions of a family of homogeneous linear systems

Example

Facts & Assumptions

Verification

Sources