Let $C$ be a smooth curve of degree $d$ in $\mathbb{P}^2$ over $\mathbb{C}$. Say $C$ is defined by $p(x,y,z)=0$, with $p$ a homogeneous degree $d$ polynomial.


In vector calculus one learns that the gradient of $p$ is normal to $C$ at every point of the curve.


In algebraic geometry, the invertible sheaf associated to the normal bundle $N\_{C|\mathbb{P}^2}$ to $C$ in $\mathbb{P}^2$, is given by $\mathcal{O}\_{\mathbb{P}^2}(d) \_{|C}$.


Is there any relationship between the gradient and the bundle or the sheaf? 



Normal bundle to a curve in P^2
Yes, there is a strong relationship between the two.


First, let's work locally in affine space rather than in projective space (it makes more
sense to work locally just because we are dealing with a sheaf, which is defined locally).
So I will consider a non-homogen


Working without a metric (as one does in at least the algebraic aspects of algebraic geometry),
it is perhaps better to talk not about the gradient of $f$, but its exterior derivative
$df$, given by the same formula: $df = f\_x dx + f\_y dy.$ Since this is differential form
valued, we will compare it with the conormal bundle to the curve $C$ cut out by $f = 0$.


Now the exterior derivative can be thought of simply as taking the leading (i.e. linear) term of $f$.


On the other hand, if $\mathcal I$ is the ideal sheaf cutting out the curve $C$, then the 
conormal bundle is $\mathcal I/\mathcal I^2$. (If $f$ is degree $d$, then $\mathcal I = \mathcal O(-d)$, and so this can be rewritten as $\mathcal O(-d)\\_{| C}$, dual to the normal
bundle $\mathcal O(d)\\_{| C}$.) Now $f$ is a section of $\mathcal I/\mathcal I^2$ (over the affine patch on which we are working), so we may certainly regard it as a section of $\mathcal I/\mathcal I^2$; this section
*is* the (image in the conormal bundle to $C$ of) the exterior derivative of $f$.


The formula $\mathcal I/\mathcal I^2$ for the conormal bundle is thus simply a structural
interpretation of the idea that we compute the normal to the curve by taking the leading term
of an equation for the curve.


