Bombieri–Lang conjecture

The equation $x+y=1$ has fraction-valued solutions such as $(1/3,2/3)$: all of its solutions form a line. More complicated polynomial equations define curves, surfaces and higher-dimensional shapes. Can the geometry of a shape force its fraction-valued solutions into a smaller part of it?

Bombieri–Lang predicts this for a particular geometric class called general type. For smooth projective curves, this class consists of those whose complex points form a surface with at least two handles; the line above is outside the class. The question here concerns positive-dimensional projective shapes, with their points at infinity included, over a number field: the rationals or a finite extension of them. Every solution with coordinates in that fixed field should lie in one proper algebraic subset, defined by additional polynomial conditions.[1] [2]

When polynomial equations define a complicated shape, must all solutions with coordinates in a fixed number field lie in a smaller algebraic part of it? That is the weak Bombieri–Lang question. A number field is the rationals with finitely many algebraic ingredients added. For the rationals themselves, $(1/3,2/3)$ solves $x+y=1$, and choosing any rational x gives another point on this line. This elementary line is not in the conjecture's general-type class.

The target is a positive-dimensional projective variety X of general type over a number field k. Projective means that the algebraic shape is considered with its points at infinity included. General type is a geometric classification; for smooth projective curves it means genus at least two. Their complex points form a surface with at least two handles. In higher dimensions the classification uses differential forms, rather than a direct count of handles.

The conjecture requires $$\overline{X(k)}^{\mathrm{Zar}}\subsetneq X.$$ The points are not Zariski dense. Here $X(k)$ is the set of points with coordinates in k; the bar means the smallest closed algebraic subset containing them. Thus an extra polynomial condition must confine all of these points without holding throughout X. This is not ordinary distance-based density. A curve inside a surface could contain infinitely many points and still be a proper algebraic subset; that illustrates the conclusion, without claiming any arbitrary pictured surface is of general type.[1] [2]

Curves and special higher-dimensional cases do not settle the general claim.[1]

Strong finiteness and geometric function-field formulations have different hypotheses and conclusions. Their results do not establish this general number-field target.[1]

Fraction-valued solutions are arithmetic objects, yet the full shape of an equation's solutions can restrict where they occur. For curves of general type, a theorem already makes those solutions finite over a fixed number field. The higher-dimensional question asks how far that link between geometry and arithmetic extends, where infinitely many solutions may still fit inside a smaller curve or surface.[1] [2]

For curves, the geometry-to-arithmetic link is already a theorem: a smooth projective curve of genus at least two has finitely many points over a fixed number field. The weak higher-dimensional conjecture extends this organizing idea while allowing exceptional smaller subsets to carry infinitely many points.

The distinction matters when studying whole families of equations. Geometry describes the complete solution space, including non-rational points, while arithmetic selects the points available over the chosen field. Confinement would connect those two descriptions through a broad geometric classification, rather than through an individual search for solutions. Stronger finiteness formulations ask for further control after exceptional subsets are removed; this page keeps the weaker target.[1] [2]