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Let $X$ be a smooth complex projective variety of dimension $n$. Consider a flag ideal $I$ on $X\times \mathbb{P}^1$, namely, $$ I=I_0+I_1t+I_2t^2+\cdots+I_{N-1}t^{N-1}+(t^N)\,, $$ where $t$ is the variable on $\mathbb{C}\subset \mathbb{P}^1$, $I_j$ are coherent ideal sheaves on $X$.

We assume that the flag ideal $I$ is special in the sense that each $I_j$ is reflexive, corresponding to an effective divisor $D_j$. Moreover, we assume that each $D_j$ is snc.

Now let $Y$ be the normalised blowing-up of $X\times \mathbb{P}^1$ along $I$. Then how can we compute the relative canonical class $K_{Y/X\times\mathbb{P}^1}$ in terms of all these $D_j$?

Even in the simplest case of deformation to the normal cone (i.e. when $N=1$), this does not seem to be clear to me.

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1 Answer 1

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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
```
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