Ideal sheaf of $Z$ twisted by $Z$

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Let $X$ be a nonsingular projective surface and $Z$ a 1-dimensional subscheme (i.e. a curve) and $m:= c_2(I_Z)$.

  1. What (geometrically) is $m$? What is the intuition behind it?
  2. What is $I_Z(Z)$? In particular, is this $0$-dimensional?