Let $C$ be a projective connected (reducible) curve over an algeraically closed field with nodes as singularities and $X=\mathbb P(\mathcal E)$ a projective bundle over $C$ (we know a desingularization of $X$). Is there a Riemann-Roch formula for vector bundles on $X$?
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Let $C_i$ be the components of $C$ and $X_i$ the corresponding components of $X$. Let $P_j$ be the intersection points of $C_i$ and $L_j$ the corresponding components of self-intersection of $X$. For any vector bundle $V$ on $X$ there is an exact sequence $$ 0 \to V \to \bigoplus V\vert_{X_i} \to \bigoplus V\vert_{L_j} \to 0 $$ (with the second map induced by the restrictions). It allows computing the Euler characteristic of $V$ (and also its cohomology spaces) in terms of the restrictions of $V$ to $X_i$ and $L_j$.
