Flat families and Hilbert polynomials

A morphism X→S is flat at x when the local ring at x is flat as a module over the local ring of S at its image. Tensoring with a flat module preserves injections and hence all short exact sequences. Flatness is not a statement that the plotted fibers look smooth, have the same number of points, or are isomorphic. It compares how relations behave under every relevant base module, making it stronger than a check at one selected residue field.

Over a principal ideal domain, a module is flat precisely when it is torsion-free, but this useful criterion does not hold unchanged over every base ring. In the example over k[t], an explicit free-module description proves flatness directly. A nonzero element killed by t instead obstructs flatness, because multiplication by t is injective on the base and tensoring would fail to preserve that injection. On a nonreduced base, even a constant collection of pointwise numerical invariants can conceal such failures.

Take a projective morphism X→S of finite presentation with S Noetherian, choose a relatively ample line bundle L, and let F be coherent and flat over S. For each point s, define P_s(n) as the Euler characteristic of F_s tensor L_s to the nth power. Over the residue field this is a polynomial in n, and for sufficiently large n it equals the dimension of global sections. The vanishing of higher cohomology in that range explains why a graded section count can recover the polynomial.

Under these hypotheses the fiber Hilbert polynomial is locally constant on S, hence constant when S is connected. One way to see the numerical constancy is to apply proper finite-presentation flat Euler-characteristic constancy to each fixed twist, then identify the resulting polynomial coefficients. For F=O_X, flatness of the morphism supplies the required flat sheaf. The polarization is part of the data: changing L changes the polynomial. We assert this forward implication, not a general converse based only on matching a few fiber values.

A plane conic can degenerate from a smooth curve to two intersecting lines while retaining its degree and arithmetic genus. The homogeneous relation must remain present throughout the family, and flatness ensures that no hidden base torsion creates or removes graded information unexpectedly. The Hilbert polynomial of a degree-two plane hypersurface is 2n+1. Its constant term gives arithmetic genus zero for a projective curve, which should not be confused with the number of irreducible components or the genus of each normalization component.

In contrast, an equation multiplied by the base parameter may disappear entirely in a special fiber. A projective embedding alone does not prevent this: properness supplies limits but does not force flatness. Computing a torsion element in one affine chart can reveal the failure, and the fiber Hilbert polynomials can then differ. In research problems this distinction is central to constructing moduli spaces: first fix the precise family and polarization, then justify the flatness and finiteness conditions before using numerical constancy as a constraint.