I'm reading Katz "Enumerative Geometry and String Theory". He shows that a degree $d$ hypersurface has cohomology class $dH $(where $H$ is the class of a hyperplane) by the following argument (p.80, I'm paraphrasing a bit):
...we can continuously deform a degree d hypersurface (given by F=0) to a union of d >hyperplanes by the equations: $$ G_t := tF(x) + (1-t) \Pi_{i=1}^{d} l_i (x) = 0 $$ where $ l_i $ are homogeneous linear forms.
I've studied the topic of intersection theory a bit already, so my question isn't about the result necessarily, but more about his argument. Specifically, what does he mean by continuous here (i.e. continuous in what topology)? For example, if we are in the the plane, and we take $d=2$, he is saying we can continuously deform a circle into a pair of lines. I'm pretty sure that deformation isn't continuous, though (or maybe it is in the complex plane?)
When $t=0$, you see that $G_0$ is the union of $d$ hyperplanes, whereas when $t=1$, we have $G_1 = F$. In the case of $\mathbb{R}$ or $\mathbb{C}$, this is actually a continuous (in the usual topology) deformation as $t$ varies continuously from 0 to 1, if we work in the projective plane.
If it seems strange that a circle could be deformed into a pair of lines, try graphing this specific instance over the real numbers for particular values of $t$: $$ t (x^2 + y^2 - 1) + (1-t)(x^2-y^2) = 0 $$
When $t=1$ this is a circle $x^2 + y^2 = 1$. When $t=0$ this is a pair of lines ($y=\pm x$).
In between, when $1/2 < t < 1$, this is an ellipse. As $t$ approaches 1/2, the ellipse becomes more and more elongated. At $t=1/2$, we are left with a pair of distinct vertical lines ($x = \pm 1/\sqrt{2}$). The "top and bottom points" of the ellipse have gone off to infinity while the "left and right points" on the x-axis approach $\pm 1/\sqrt{2}$. Then for $0 < t < 1/2$, the two lines branch into a hyperbola. At $t=0$, the hyperbola has degenerated into a pair of distinct lines.
In the affine plane, we seem to have encountered a discontinuity at $t=1/2$. But in the projective plane, all these conics look the same. The "points at infinity" are actual points in the projective plane, and the two "branches" of the hyperbola join up twice at infinity (once for each of the two asymptotes) to make a single connected component, just like an ellipse.