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    Fibre dimension and upper semicontinuity

    T-MORFIBDIM

    Fibre dimensions can jump only up on special loci: the general fibre has dimension $\dim X-\dim Y$, and larger-fibre loci are closed.Suggest a slogan

    (Noether normalization (affine))

    Let \(k\) be an infinite field and \(X \subseteq \AA^{n}_k\) an affine variety of dimension \(d\). Then there exists a finite morphism \(X \to \AA^{d}\) which is the restriction to \(X\) of a linear map \(\AA^{n} \to \AA^{d}\).

    (Noether normalization (projective))

    Let \(k\) be an infinite field and \(X \subseteq \PP^{n}_k\) a projective variety of dimension \(d\). Then there exists a finite morphism \(X \to \PP^{d}\) which is the restriction to \(X\) of a linear projection \(\PP^{n} \dashrightarrow \PP^{d}\) with centre a linear subspace \(\PP^{k} \subseteq \PP^{n}\) disjoint from \(X\), where \(k + d = n - 1\).

    The linear forms are chosen from a nonempty Zariski open subset of a space of linear maps, which has \(k\)-points because \(k\) is infinite. Over a finite field a linear map need not exist: over \(\FF_2\), no linear map \(\AA^2 \to \AA^1\) restricts to a finite morphism on the irreducible cubic \(V(xy(x+y) + 1)\). Noether normalization still holds over every field if nonlinear maps such as \((x_1 - x_n^{e_1}, \ldots, x_{n-1} - x_n^{e_{n-1}})\) are allowed.

    (Fibre dimension)

    Let \(f : X \to Y\) be a dominant morphism of integral schemes of finite type over a field, and set \(e = \dim X - \dim Y\). Then every irreducible component of every nonempty fibre \(X_y\) has dimension at least \(e\), and there is a nonempty open \(V \subseteq Y\) with \(\dim X_y = e\) for every \(y \in V\).

    (Semicontinuity)

    The function \(x \mapsto \dim_x X_{f(x)}\) is upper semicontinuous on \(X\): the locus where the fibre dimension is at least \(n\) is closed. If \(f\) is flat then \(\dim_x X_{f(x)} = \dim_x X - \dim_{f(x)} Y\) at every point, so the fibre dimension is locally constant.

    Upper semicontinuity means fibre dimension can jump up on special loci, never down. The blowup is the model: fibre dimension \(0\) generically, \(1\) over the origin, and the jump locus \(\ts{0}\) is closed. It cannot go the other way, because a component of a fibre is cut out by \(\dim Y\) equations locally and Krull’s height theorem bounds the drop.

    The two halves are used differently. The lower bound proves positive-dimensional fibres in dimension-comparison arguments. The generic equality gives the dimension formula \(\dim X = \dim Y + \dim(\text{general fibre})\).

    Flatness removes this jumping by forcing the expected local fibre dimension everywhere.

    State the fibre dimension theorem.
    In what sense is fibre dimension semicontinuous, and in which direction?
    What is Noether normalization?

    Dependencies

    • D-MORFIBThe fibre of a morphism as a base changeuses

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