Normal cone

normal cones in real vector spaces
In algebraic geometry, the normal cone of a subscheme of a scheme is a scheme analogous to the normal bundle or tubular neighborhood in differential geometry.

For a Cartesian square of schemes with the vertical map, there is a closed embedding of normal cones.

Let be a scheme of finite type over a field and a closed subscheme. If is of pure dimension r; i. e., every irreducible component has dimension r, then is also of pure dimension r. (This can be seen as a consequence of #Deformation to the normal cone.) This property is a key to an application in intersection theory: given a pair of closed subschemes in some ambient space, while the scheme theoretic intersection has irreducible components of various dimensions, depending delicately on the positions of , the normal cone to is of pure dimension.

Let be an effective Cartier divisor. Then the normal bundle to it (or equivalently the normal cone to it) is

The normal cone's geometry can be further explored by looking at the fibers for various closed points of Note that geometrically is the union of the plane with the axis , so the points of interest are smooth points on the plane, smooth points on the axis, and the point on their intersection. Any smooth point on the plane is given by a map for and either or Since it's arbitrary which point we take, for convenience let's assume Hence the fiber of at the point is isomorphic to giving the normal cone as a one dimensional line, as expected. For a point on the axis, this is given by a map hence the fiber at the point is which gives a plane. At the origin , the normal cone over that point is again isomorphic to .

For the nodal cubic curve given by the polynomial over , and the point at the node, the cone has the isomorphism showing the normal cone has more components than the scheme it lies over.

Suppose is an embedding. This can be deformed to the embedding of inside the normal cone (as the zero section) in the following sense: to is the quotient stack which is the stack of fppf torsors on A concrete interpretation of this stack quotient can be given by looking at its behavior locally in the etale topos of the stack .

More concretely, suppose there is an étale morphism from an affine finite type scheme together with a locally closed immersion into a smooth affine finite type scheme Then one can show meaning we can understand the intrinsic normal bundle as a stacky incarnation for the failure of the normal sequence to be exact on the right hand side. Moreover, for special cases discussed below, we are now considering the quotient as a continuation of the previous sequence as a triangle in some triangulated category. This is because the local stack quotient can be interpreted as in certain cases.

The intrinsic normal cone to , denoted as , is then defined by replacing the normal bundle with the normal cone ; i. e.,

Example: One has that is a local complete intersection if and only if In particular, if is smooth, then is the classifying stack of the tangent bundle , which is a commutative group scheme over
More generally, let is a Deligne Mumford Type (DM type) morphism of Artin Stacks which is locally of finite type. Then is characterised as the closed substack such that, for any étale map for which factors through some smooth map (e. g., ), the pullback is: