I know there is a spectral sequence for a variety with normal crossing singularities $X$ which gives a tool for making the computation of the mixed hodge structure computable. How can I compute the mixed hodge structure for a projective variety which has nonnormal crossing singularities? For example, if I have $$ X = \text{Proj}\left( \frac{\mathbb{C}[x,y,z]}{xy(x+y)} \right) $$ how can I compute its Mixed Hodge Structure?
1 Answer
Let me expand my previous comment, and place it in a more general context. Suppose that $X$ is an singular connected but possibly reducible projective curve. Let $\tilde X\to X$ be the normalization. $\tilde X$ is a disjoint union of the normalizations of the irreducible components of $X$. Let us say that there $n$ such components. Then it's not difficult to see that $H^0(X) = \mathbb{Z}$ and $H^2(X)= H^2(\tilde X)=\mathbb{Z}(1)^n$ as MHS. For $H^1$, we have an exact sequence $$0\to W_0\to H^1(X)\to H^1(\tilde X)\to 0$$ In practice, this is pretty easy to compute. In your example, $H^1=0$ as I said in my comment.

$\begingroup$ Oh cool! So if I have two elliptic curves intersecting generically in $\mathbb{P}^2$ then I have eight additional weight 0 cycles. $\endgroup$ Aug 9, 2017 at 19:54