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Let $X_s = \bigcap_{i=1}^s H_i \subset \mathbb{P}^N$ be a complete intersection, where each $H_i$ is a hypersurface of degree $d_i$.

Is $X_{s-1}$ also a complete intersection? Is the conormal sheaf of $X_s$ in $X_{s-1}$ isomorphic to $\mathcal{O}_{X_s}(-d_s)$?

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Yes for the first question. For the second replace $d_i$ with $d_s$, then the answer is positive as well. –  Sasha Dec 14 '12 at 7:33
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