In the case $N=1$, where the ideal is $I = \mathcal{O}(-D) + (t)$ for some snc divisor $D$ on $X$, $I$ is a global complete intersection of codimension 2 ($V(I) = D \times \{0\}$), with normal bundle $\mathcal{N} = \mathcal{O}(D)|_D \oplus \mathcal{O}_D$. If $Y := \mathrm{Bl}_I X$ and if $\pi\colon Y \to X$ is the projection, then the exceptional locus $E = \pi^{-1}(D \times \{0\})$ can be identified as $E \simeq \mathrm{Proj}_D \mathcal{N}$, a $\mathbb{P}^1$-bundle over $D \times \{0\}$. In particular, if $D = \sum_i D_i$ (with the $D_i$ smooth and irreducible) then $E = \sum_i E_i$ where the $E_i = \pi^{-1}(D_i)$ are also smooth and irreducible.

Now, we know that $K_Y - \pi^* K_{X \times \mathbb{P}^1} = \sum_i a_i E_i$. The coefficient $a_i$ only depends on the valuation corresponding to $E_i$ (see Rmk. 2.23 of Kollár-Mori's *Birational Geometry of Algebraic Varieties*), so it can be computed on any neighborhood of the generic point of $E_i$. This essentially reduces to the case where $D$ is smooth, and so Lem. 2.29 of the above reference shows $a_i=1$ for all $i$.

The above argument fails when $N>1$ since $I$ is no longer a complete intersection. However, if we assume that $I_i = \mathcal{O}(-D_i)$ for each $i$, where the $D_i$ are *smooth* divisors on $X$ so that $\sum_i D_i$ is snc, there might be a tractible description of $E$ in terms of the *strata* of $\sum D_i$ (the components of intersections $\cap_{i \in I} D_i$, where $I \subseteq \{0, \dots, N-1\}$). For example, if $X = \mathbb{A}^2$ and $I = (x) + (yt) + (t^2)$, the exceptional locus has 2 components: a $\mathbb{P}^1$-bundle over $V(x) \times \{0\}$ and a divisor centered over $(0, 0)$. At the risk of extrapolating from 1 example: one could ask if with the assumptions of this paragraph, and setting $Z_m = \bigcap_{i=0}^m D_i$, whether $K_Y - \pi^* K_X = \sum_{m=0}^{N-1} (m+1) E_m$, where $E_m$ is an exceptional divisor centered at $Z_m$. Note that $m+1 = \mathrm{codim}(Z_m \subset X \times \mathbb{P}^1) -1$.

*Edit*: Macaulay2 code for the example.

```
i1 : k = ZZ/9973
o1 = k
o1 : QuotientRing
i2 : XtimesP1 = k[x, y, t]
o2 = XtimesP1
o2 : PolynomialRing
i3 : I = ideal(x, y*t, t^2)
2
o3 = ideal (x, y*t, t )
o3 : Ideal of XtimesP1
i4 : Y = reesAlgebra(I)
o4 = Y
o4 : QuotientRing
i5 : E = I*Y
2
o5 = ideal (x, y*t, t )
o5 : Ideal of Y
i6 : primaryDecomposition E
2 2
o6 = {ideal (t, x, w ), ideal (x, t , y*t, y )}
1
o6 : List
i7 : describe Y
XtimesP1[w , w , w ]
0 1 2
o7 = -----------------------------------------
2
(y*w - t*w , t w - x*w , y*t*w - x*w )
1 2 0 1 0 2
```
```