Tangent cones

But e(y) = dim0 Ty = r, by assumption, so this is a contradiction. This proves that ij) is an isomorphism, hence P v is free.  

Problem. Let X be an irreducible noetherian scheme all of whose stalks ox at closed points are principal valuation rings. If T is a coherent ox-module, show that 

We are ready to go back to geometry. Let A be an algebraically closed field, let X be a scheme of finite type over k, and let a be a closed point of X. The scheme X has a tangent cone at x defined as follows  

A priori, it might look as though this definition depended on the particular embedding of a neighbourhood of x in affine space. We can easily make it intrinsic through  

Definition 2. Let O be a local ring, with maximal ideal m. Then 

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Therefore, the tangent cone is a union of lines through the origin taken with multiplicities. 

The ideal on the right is nothing but the ideal defining the ith affine piece of the projectivized tangent cone. ... 

Corollary. If X is an r-dimensional variety, then the tangent cone to X at x is pure r-dimensional. In general... 

Note that the subscheme of Bx(X) defined by x 0 is the point P doubled since Bx(X) is tangent to the exceptional curve E at P. And, indeed, the tangent cone in this case is a double line, as (V) requires. 

From a technical point of view, the big drawback about the tangent cone is that it is non-linear. It is always easier to handle essentially linear objects. The most natural way around this is to study the linear hull of the tangent cone, which we will call the tangent space. Let X be a scheme of finite type over At, and let x X be a closed point. 

We can even embed the tangent cone inside the tangent space in a completely intrinsic way. There is a canonical surjection ... 

Once again, we see that Fxcx is exactly a linear hull of the tangent cone. 

This intuitively reasonable definition, like the provisional ones we made for the tangent cone and tangent space, is not really intrinsic. 

If C is not hyperelliptic, it turns out that the tangent cone Tv,o to V at 0 G Jac is just a cone over the curve C itself more precisely, if P is the projective space of 1-dimensional subspaces of TjaC)o> then the canonical curve <2>(X) sits in P, and... 

The study of tangent cones at singular points of the theta-divisor has been carried out by M. Green in [G] where he proves the following theorem For... 

Qijdx dx = 0, namely the tangent cone through the origin, and a hyperplane, the polarplane of the origin, are then identified at the same time in each tangent space. 

The polar plane contains the contact point of the tangent cone with the surface. The polar hyperplane should represent the electromagnetic potentials and the cone, alternatively, the gravitational potentials (Fig. 3). 

The equation of the tangent cone with its vertex at the origin is... 

The contact points of the tangent cones with the quadric are known to lie in the polar hyperplane. 

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