# Relative canonical class of blowing-up a flag ideal

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.

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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