blowing-up and tangent cone: essentially identical concepts?

Question

After reading about the concepts of blowing-up and tangent-cone of a curve at a point $P$, i have the following understanding: The blowing-up gives us the slopes of the tangent(s) of the curve at point $P$, while the tangent cone gives us the tangent line(s) themselves. So it seems that the two concepts coincide. So why do we need both of them? Is this understanding correct?

Answer 1

The blowing-up at one point P by which another curve passes (I suppose you're dealing with plane curves) does contain in its exceptional divisor all directions from P, and in the case where the 2nd curve is, e.g.smooth at P and does not pass through the tangent cone at P, it will yield different points at the exceptional divisor through P.

The exceptional divisor contains much more info than just "the tangent cone".

Consider the curve

$$C \equiv y^2=x^3+x^2,$$

whose tangent cone at $(0,0)$ consists of $y^2-x^2=0.$ The tangent cone will yield exactly two points, $(1:1), (1:-1)$ at the exceptional divisor, and if you blow up the plane you'll get a ${\mathbb P}^1$ which sends two points to $P$ corresponding to the two tangents. This curve $D$ is called the strict transform of $C$, and it is the closure of $C-P$ at the blown-up variety.

This is well explained in: Shafarevich, Basic Algebraic Geometry.

LATER ADDITION: One can iterate blow-ups, but cannot iterate tangent cones. Suitable composition of blow-ups (aka monoidal transformations) will reveal the behaviour of $C$ "infinitely close to $p$".

LINKING BLOW-UPS AND THE TANGENT CONE: (References: Beauville's Complex Algebraic Surfaces, which is shorter and handier, or Hartshorne's AG, Chap. V)

For a plane curve $C$, multiplicity can be obtained as $m_p(C)=\mbox{inf}_{\ell} \mu_p(C\bullet \ell)$, i.e. the minimum multiplicity at $p$ of intersection of $C$ with a line $\ell$. Replacing $\ell$ with a curve $D$ smooth at $p$ you obtain the multiplicity of a curve $C$ on a smooth surface $X$ at $p\in C$ -- this will become handy below.

The tangent cone consists of the lines where this degree is strictly bigger, say $\ell_1, \cdots , \ell_r.$

After performing a blow-up, all lines $\ell_i$ lift to $D_i\equiv {\mathbb P}^1$, and cut the exceptional divisor $E$ precisely at one point $Q_i$. So far, so good.

If the singularity of $C$ at $P$ is ordinary, then you have smooth points $Q_i\in E$ of the strict transform ${\hat C}$, and you are done: you desingularized $C$ around $p$. The tangent cones of $Q_i$ will simply consist of a tangent line after a suitable imbedding of the blown-up surface in a projective space: thus they are points in ${\mathbb P}(T_{Q_i}{\hat X})$.

However, it may well be that the points $Q_i$ are still singular, in which case one needs further work. ${\hat C}\bullet E=\sum a_i Q_i, \sum a_i=m_p(C),$ and this shows that $m_{Q_i}({\hat C})\leq a_i < m_p(C).$ One need continue blowing up everyone until all points have multiplicity one.

EXAMPLE: Take the origin and the following irreducible affine curve:

$$f(x,y)= x^3(x+y)^2y^4+(x-y)^{10}+g(x,y)=0$$

(you might need to add terms of degree $\geq 11$ to make the example more interesting; if $g=0$, the present example is irreducible).

You will have $3$ points in the exceptional divisor with non-trivial tangent cones. The points marked at the exceptional divisor do not offer more information than the tangent cone, but the interaction of $E$ with the strict transform ${\hat C}$ allows for the process to be done on the blown-up surface until one desingularises the curve.

Again you will find in the references mentioned that $$p_a({\hat C})=p_a(C)-\frac{m_p(m_p-1)}{2},$$ and since $p_a(D)=\dim H^1({\mathcal O}_D)\geq 0$ this process will eventually stop when the multiplicity of every point equals $1$ (examine the additional term above).

This is what we called \emph{embedded desingularisation}, and thus $E$ is important due to its role in the blown-up surface, though if you forget all this and just think of ${\hat C} \cap E$ as a mere set of points, you will be missing this picture completely.

Note that even the multiplicity of ${\hat C}\bullet E$ at the points which come from the tangent cone is important for further study (and coincides with the multiplicity of the corresp. line at the tangent cone).

More complicated examples are to be found in any book where the Puiseux expansion is treated, e.g. Walker's Algebraic Curves, or the book by E. Casas-Alvero.