ag.algebraic geometry - 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.