Are there any nice descriptions of the Picard group of a non-reduced double structure on $\mathbb{P}^1?$ In particular, I'm looking for a description that makes clear the difference between the Picard group of the double line $x^2=0$ in $\mathbb{P}^2$ (in my mind the Picard group should be zero-dimensional, since the curve is cut out by a quadratic equation and thus "genus zero") and the Picard group of the double structure $f(x,y,z)^2=0,$ where $f$ is a smooth conic (which should be three dimensional, since the curve is genus three).
Picard group of double “line”
algebraic-geometry
Best Answer
Let $X$ be a variety and $Y$ a multiple structure on $X$. Then, you have an exact sequence, $0\to I\to O_Y\to O_X\to 0$. If $I^2=0$, one has an exact sequence, $0\to I\to O_Y^*\to O_X^*\to 0$, where the first map is $t\mapsto 1+t$. If $X$ is projective, one has $H^0(O_X^*)=k^*$ and then, one has long exact sequence $0\to H^1(I)\to \operatorname{Pic} Y\to \operatorname{Pic} X\to H^2(I)$.
In your case of curves, $H^2(I)=0$ and the rest of the calculations are straightforward.