Blow-up Diagonal Depth and Wonderful Compactifications

1 Introduction

Let $X$ be a smooth proper variety over an algebraically closed field. The diagonal kernel

controls the identity functor of $D^b_{\mathrm{coh}}(X)$. A resolution of this diagonal by exterior product kernels

gives an upper bound for the Rouquier dimension of $D^b_{\mathrm{coh}}(X)$. This principle is part of the general theory of diagonal dimension.

The purpose of this paper is not to introduce a new diagonal invariant. Instead, we use a resolved dg-enhanced model of diagonal dimension, called coherent product diagonal depth, to prove a new stability theorem under blow-ups.

The guiding question is:

The answer is dimension-sharp.

Theorem 1.1: blow-up theorem.

Let $X$ be smooth proper and let $Z\subset X$ be a smooth closed subvariety of codimension $c$. Let

be the blow-up. Then

Equivalently, for

one has

Thus a smooth blow-up does not create new diagonal-depth defect beyond the defect already present in the ambient variety and the center; in fact, the contribution of the center is shifted down by one.

The proof uses Orlov's blow-up decomposition

where $E=\mathbb P_Z(N_{Z/X})$ is the exceptional divisor. We do not apply an additive semiorthogonal-decomposition bound. Instead, we construct a single filtered Fourier--Mukai kernel for the identity. The base part is pulled back from the diagonal of $X$. The exceptional part is resolved by a finite relative bar complex over the $c-1$ exceptional blocks. Each bar term is controlled by the diagonal of $Z$, and the bar length contributes exactly $c-2$.

The blow-up theorem iterates over clean building sets.

Theorem 1.2: wonderful compactifications.

Let $X$ be smooth proper, let $\mathcal G$ be a clean building set of smooth proper subvarieties of $X$, and let $X_{\mathcal G}$ be the corresponding wonderful model. Let $\mathcal S(\mathcal G)$ be the induced clean arrangement, with $X$ included as the empty stratum. Then

Equivalently,

In particular, if all strata are diagonal-depth optimal, then $X_{\mathcal G}$ is diagonal-depth optimal.

Our first main application is Kapranov's model of $\overline M_{0,n}$. Kapranov realizes $\overline M_{0,n}$ as an iterated blow-up of $\mathbb P^{n-3}$ along linear subspaces and their transforms. Since projective spaces are diagonal-depth optimal, Theorem 1.2 gives

Rouquier's lower bound gives

Since

we obtain the following equality.

Theorem 1.3.

For all $n\ge3$,

This proof is non-equivariant and does not use the even Hassett-space diagonal-depth analysis appearing in the Castravet--Tevelev approach. That approach remains important for symmetric and equivariant refinements, but the Rouquier-dimension equality above follows directly from Kapranov's blow-up model and Theorem 1.1.

We also prove:

Theorem 1.4.

If $X$ is smooth proper and diagonal-depth optimal, then its Fulton--MacPherson compactification $X[n]$ is diagonal-depth optimal. Hence

Finally, in characteristic zero, Theorem 1.1 implies that diagonal-depth optimality is a birational invariant for smooth projective varieties of dimension at most three. Hence every smooth projective rational variety of dimension at most three is diagonal-depth optimal and satisfies Orlov's expected equality for Rouquier dimension.

2 Coherent product diagonal depth

Throughout, $k$ is an algebraically closed field. Unless otherwise stated, all varieties are smooth and proper over $k$, and all functors are derived.

We fix dg enhancements of the bounded derived categories of coherent sheaves. This allows us to work with filtered twisted complexes of Fourier--Mukai kernels.

Definition 2.1.

Let $X$ be smooth proper. A product kernel on $X\times X$ is an object of the form

A finite direct sum of product kernels is again called a product kernel.

For

we write

if $K$ is represented in the dg enhancement by a finite twisted complex

such that every $P^r$ is a product kernel.

The coherent product diagonal depth of $X$ is

The normalization is such that

Remark 2.2.

This is a resolved model of diagonal dimension. If $\mathcal O_{\Delta_X}$ is generated in $d$ steps by a single product object $A\boxtimes B$, then it admits a product twisted-complex presentation of amplitude $d$. Conversely, a finite product twisted complex can be generated by the single product object

after allowing finite sums and direct summands. Thus the invariant agrees with diagonal dimension under the usual enhanced and idempotent-complete conventions. We use the notation $\operatorname{CPD}$ to keep track of the resolved product-complex model needed for the blow-up proof.

Lemma 2.3: pullback.

Let $g:Y\to X$ be a morphism between smooth proper varieties. Then

for every

Proof. It suffices to check product kernels. We have

Pulling back a product twisted complex gives a product twisted complex of the same amplitude.

Lemma 2.4: pushforward on support.

Let $i:W\to Y$ be a proper morphism between smooth proper varieties. Then

for every

Proof. For product kernels,

The claim follows by applying $R(i\times i)_*$ to a product twisted complex.

Lemma 2.5: resolved cone-splicing.

Let

be a distinguished triangle in $D^b_{\mathrm{coh}}(X\times X)$. Suppose $A$ and $C$ are represented by product twisted complexes of amplitudes $a$ and $c$, respectively, and suppose the triangle is represented in the dg enhancement so that

Then

Proof. Represent $A$ by a product complex in degrees $0,\dots,a$, and $C$ by a product complex in degrees $0,\dots,c$. Then $C[-1]$ contributes terms in degrees $1,\dots,c+1$. The mapping cone has terms

and therefore lies in degrees

Each term is a finite direct sum of product kernels. This proves the claim.

Lemma 2.6: Rouquier bound.

For $X$ smooth proper,

Proof. A product kernel

acts by the Fourier--Mukai functor

Thus its image lies in the triangulated subcategory generated by $B$. A product twisted complex resolving $\mathcal O_{\Delta_X}$ gives a filtration of the identity functor with the same number of extension steps. Applying this filtration to an arbitrary object of $D^b_{\mathrm{coh}}(X)$ shows that a finite direct sum of the objects appearing as right factors in the product kernels strongly generates $D^b_{\mathrm{coh}}(X)$ in at most $\operatorname{CPD}(X)$ steps.

Lemma 2.7: products.

For smooth proper $X$ and $Y$,

Proof. The diagonal satisfies

Taking product twisted complexes for the two factors and totalizing by total degree gives a product twisted complex of amplitude the sum of the two amplitudes.

Lemma 2.8: projective spaces.

For every $m\ge0$,

Proof. Beilinson's resolution of the diagonal gives a product resolution of $\mathcal O_{\Delta_{\mathbb P^m}}$ of length $m$. Hence

The reverse inequality follows from Rouquier's lower bound and Lemma 2.6.

Lemma 2.9: projective bundles.

Let

be a projective bundle with $\operatorname{rank}\mathcal V=r$. Then

Proof. The relative diagonal of $\mathbb P_X(\mathcal V)$ over $X$ has the relative Beilinson resolution of length $r-1$. Pulling back a product twisted complex for $\mathcal O_{\Delta_X}$ and tensoring with this relative resolution gives a product twisted complex for the absolute diagonal of total amplitude at most

3 Orlov's blow-up formula and the base projector

Let

be a smooth closed subvariety of codimension $c$, and let

be the blow-up. If $c=1$, then $Y\simeq X$, and all statements below are immediate. Thus, throughout the proof of the blow-up theorem, we assume $c\ge2$.

Let

be the exceptional divisor and its projection. We use the convention

For $m=1,\dots,c-1$, define

Orlov's blow-up formula gives a semiorthogonal decomposition

Let

Then

The functor

has Fourier--Mukai kernel

The counit

gives a triangle of Fourier--Mukai kernels

Here $K_{\mathcal E}$ is the complementary projector: it kills $f^*D^b(X)$ and restricts to the identity on $\mathcal E$.

By Lemma 2.3,

Thus the blow-up theorem reduces to bounding $K_{\mathcal E}$.

4 The exceptional relative bar complex

This section proves the technical core.

Let

Choose a product twisted complex

of amplitude $d_Z$. Thus

If $V\in\operatorname{Perf}(Z)$, then

is again a product twisted complex of the same amplitude, since

Similarly, tensoring by perfect complexes pulled back from the first factor preserves product amplitude.

Hence every kernel of the form

has product depth at most $d_Z$.

Lemma 4.1: block bimodules.

For $m,n\in{1,\dots,c-1}$, let

be the bimodule between the $m$-th and $n$-th exceptional blocks. This bimodule is represented by a kernel on $Z\times Z$ of the form

for some

In particular, $\mathcal M_{mn}$ has product depth at most $d_Z$.

Proof. The functor $\Phi_m$ is

Since

the right adjoint of $\Phi_m$ is

Applying this to $H=\Phi_n(G)$, we obtain

For the divisor embedding $j:E\hookrightarrow Y$, the derived self-intersection $Lj^*j_*(-)$ is given by tensoring with the two-term Koszul complex

up to the standard cohomological convention. Therefore $\Phi_m^R\Phi_n$ is a finite extension of tensor functors

Since $p:E\to Z$ is projective, these coefficients are perfect complexes on $Z$. Hence $\Phi_m^R\Phi_n$ is represented by a diagonal kernel

for some $V_{mn}\in \operatorname{Perf}(Z)$.

Replacing $\mathcal O_{\Delta_Z}$ by $P_Z^\bullet$ gives a product twisted complex of amplitude $d_Z$. Thus $\mathcal M_{mn}$ has product depth at most $d_Z$.

Lemma 4.2: endpoint kernels.

The embedding of the $m$-th exceptional block into $D^b(Y)$ is represented by a kernel

supported on the graph of $p:E\to Z$, followed by the embedding $j:E\hookrightarrow Y$. Up to the convention of source and target factors,

The corresponding outgoing kernel representing the relevant adjoint is of the same graph type, with twist $\mathcal O_E(m-1)$ and a cohomological shift.

Consequently, if a chain of block bimodules produces a kernel

on $Z\times Z$, then inserting the endpoint kernels gives a kernel on $Y\times Y$ of the form

where $L,M$ are line bundles on $E$, $V\in\operatorname{Perf}(Z)$, and $s\in\mathbb Z$. Every kernel of the form (4.2) has product depth at most $d_Z$.

Proof. The first assertion follows directly from the Fourier--Mukai kernel of

The adjoint kernel is obtained from (4.1), hence is again supported on the same incidence correspondence, with the indicated twist and shift.

Now insert a diagonal kernel $\Delta_{Z*}V$ between these two endpoint kernels. The convolution identifies the two $Z$-coordinates and produces exactly a kernel supported on

and then pushed forward to $Y\times Y$ by $j\times j$. This gives (4.2).

Replacing $\mathcal O_{\Delta_Z}$ by $P_Z^\bullet$, a product term

becomes

Thus the product amplitude remains $d_Z$.

Lemma 4.3: directed dg model.

Choose dg enhancements of all categories involved. The exceptional category $\mathcal E$ is Morita equivalent to a finite directed dg category $\mathfrak A$ with components

The ordering is chosen so that morphisms in the forbidden semiorthogonal direction vanish. Let

be the diagonal subcategory, and let

be the strictly triangular radical bimodule.

Then

The identity bimodule of $\mathfrak A$ has the relative bar resolution

The term of bar degree $\ell$ is a direct sum over strict chains of exceptional blocks

No chain has length $\ell>c-2$.

Proof. This is the standard directed dg model associated to a semiorthogonal decomposition. The relative bar resolution over the diagonal subcategory resolves the identity bimodule. Since there are $c-1$ ordered components and $J$ is strictly triangular, there are no strict chains of length $c-1$ or more.

Lemma 4.4: kernel realization of bar terms.

Under the Fourier--Mukai realization of the Morita equivalence $\mathfrak A\simeq\mathcal E$, the bar term attached to a chain

is represented by the convolution

This convolution is a kernel of the form

where $L_\sigma,M_\sigma$ are line bundles on $E$, $V_\sigma\in\operatorname{Perf}(Z)$, and $s_\sigma\in\mathbb Z$. Hence every such chain kernel has product depth at most $d_Z$.

Proof. Each middle factor $\mathcal M_{m_i,m_{i+1}}$ is represented on $Z\times Z$ by

Convolution of diagonal kernels is diagonal:

By induction,

for

Inserting the endpoint kernels gives (4.3) by Lemma 4.2. The product-depth estimate follows by replacing $\mathcal O_{\Delta_Z}$ with the fixed product complex $P_Z^\bullet$.

Lemma 4.5: compatibility with the bar differential.

The product complexes representing the chain kernels in Lemma 4.4 may be chosen so that the bar differential has filtration degree (0) in the $P_Z^\bullet$-direction.

Proof. The same product complex

is used for every chain kernel. The data varying with the chain $\sigma$ are only

The face maps in the bar differential are induced by composition of adjacent block bimodules. On the kernel side, these maps are induced by canonical composition morphisms

and by the corresponding composition maps among endpoint line-bundle twists. These morphisms act on the coefficient factors

and leave the fixed product complex $P_Z^\bullet$ unchanged. Therefore each face map is represented by a morphism of product twisted complexes of degree (0) in the $P_Z^\bullet$-filtration.

Proposition 4.6: exceptional projector depth.

The exceptional projector kernel $K_{\mathcal E}$ satisfies

Proof. The relative bar complex of Lemma 4.3 resolves the identity bimodule of the directed dg category modeling $\mathcal E$. Under the Fourier--Mukai realization, it becomes a twisted complex resolving the exceptional projector kernel $K_{\mathcal E}$.

The bar complex has bar degree

By Lemma 4.4, each bar-degree-$\ell$ summand is represented by a product twisted complex of amplitude $d_Z$. By Lemma 4.5, the bar differential is compatible with the $P_Z^\bullet$-filtration. Thus totalizing with

gives a product twisted complex of amplitude

This proves the proposition.

5 The blow-up theorem

Theorem 5.1.

Let $X$ be smooth proper and let $Z\subset X$ be a smooth closed subvariety of codimension $c$. Let

Then

Proof. If $c=1$, the blow-up is isomorphic to $X$, so the result is immediate. Assume $c\ge2$.

Set

By (3.4),

By Proposition 4.6,

Using the triangle

and the resolved cone-splicing Lemma 2.5, we obtain

Corollary 5.2: defect form.

Let

Then

Proof. Since

Theorem 5.1 gives

Corollary 5.3.

If $X$ is CPD-optimal and $\eta(Z)\le1$, then $\operatorname{Bl}_ZX$ is CPD-optimal. In particular, if both $X$ and $Z$ are CPD-optimal, then $\operatorname{Bl}_ZX$ is CPD-optimal.

Proof. If $X$ is CPD-optimal, then

If

then Corollary 5.2 gives

Since $\operatorname{CPD}(Y)\ge\dim Y$, we have $\eta(Y)=0$.

Lemma 5.4: blow-down monotonicity.

Let

Then

Proof. Since

we have

Indeed,

Applying $R(f\times f)_*$ to a product complex for $\mathcal O_{\Delta_Y}$ gives a product complex for $\mathcal O_{\Delta_X}$, because

Thus

6 Wonderful compactifications

We recall the minimum needed from the theory of wonderful compactifications.

Let $X$ be smooth proper. A clean arrangement $\mathcal S$ is a finite collection of smooth closed subvarieties, closed under clean intersections. A building set $\mathcal G\subseteq\mathcal S$ is a subcollection such that every stratum $S\in\mathcal S$ is obtained as a transverse intersection of the minimal elements of $\mathcal G$ containing $S$.

The wonderful model $X_{\mathcal G}$ is obtained by blowing up the elements of $\mathcal G$ in an admissible order. If

is such an order, set

where $C_k\subset X_{k-1}$ is the dominant transform of $G_k$. Then

and all $C_k$ are smooth.

We use the following trace property.

Lemma 6.1: trace centers.

After blowing up $G_1,\dots,G_{k-1}$, the center $C_k$ is the wonderful model of $G_k$ for the trace building set induced by the earlier centers. Its strata are precisely the strata

such that

Moreover,

and for $S\subseteq G_k$,

Proof. This is the standard compatibility of dominant transforms with clean restriction in the construction of wonderful models. The admissibility of the order guarantees that the earlier centers restrict to a building set on $G_k$, and cleanliness gives additivity of codimension.

Theorem 6.2: wonderful defect bound.

Let $X$ be smooth proper and let $\mathcal G$ be a clean building set with induced arrangement $\mathcal S(\mathcal G)$, including $X$. Then

Equivalently,

Proof. Let

We prove by induction over an admissible blow-up order that

For $k=0$, this is true by the definition of $M$.

Assume it for $X_{k-1}$. By Theorem 5.1,

It remains to bound the second term.

By Lemma 6.1, $C_k$ is the wonderful model of $G_k$ for the trace building set. We apply the present theorem by induction on the pair

ordered lexicographically. Since $G_k$ has smaller dimension than $X$, the induction is well-founded. We get

Adding

gives two types of terms.

First,

Second, for $S\subsetneq G_k$,

Thus

Therefore

At the final step, $X_N=X_{\mathcal G}$, so the theorem follows.

Corollary 6.3.

If every stratum $S\in\mathcal S(\mathcal G)$ is CPD-optimal, then $X_{\mathcal G}$ is CPD-optimal.

Proof. For $S=X$, the contribution is

For $S\ne X$,

Thus Theorem 6.2 gives

Since

and

equality follows.

7 Linear wonderful models and moduli spaces of pointed curves

Proposition 7.1: linear wonderful models.

Let $X=\mathbb P^N$, and let $\mathcal G$ be a building set of linear subspaces. Then the wonderful model $X_{\mathcal G}$ is CPD-optimal:

Proof. Every stratum of a linear arrangement is a projective space. By Lemma 2.8, projective spaces are CPD-optimal. The result follows from Corollary 6.3.

\textbf{Theorem 7.2: Rouquier dimension of $\overline M_{0,n}$.}

For all $n\ge3$,

Proof. Kapranov realizes $\overline M_{0,n}$ as an iterated blow-up of

along the linear spans of subsets of $n-1$ points in linear general position, blown up in increasing order of dimension. Equivalently, this is the wonderful model of the corresponding linear arrangement.

By Proposition 7.1,

By Lemma 2.6,

Rouquier's lower bound gives

Thus equality holds.

Remark 7.3: relation with the Castravet--Tevelev program.

Castravet and Tevelev construct full $S_n$-invariant exceptional collections on $\overline M_{0,n}$. Their construction is crucial for equivariant and representation-theoretic refinements. The proof above is different: it is non-equivariant and uses only Kapranov's blow-up model together with Theorem 5.1. In particular, it does not rely on diagonal-depth optimality for the even Hassett blocks appearing in the Castravet--Tevelev reduction.

8 Fulton--MacPherson compactifications

Let $X$ be smooth proper of dimension $d$. The Fulton--MacPherson compactification $X[n]$ is the wonderful compactification of $X^n$ associated to the building set of diagonals

The strata of the polydiagonal arrangement are products $X^r$, corresponding to partitions of ${1,\dots,n}$ into $r$ blocks.

Theorem 8.1.

For every smooth proper $X$,

If $X$ is CPD-optimal, then $X[n]$ is CPD-optimal. Consequently,

Proof. Let

By Lemma 2.7,

A proper polydiagonal stratum is isomorphic to $X^r$ for some $1\le r<n$. Its codimension in $X^n$ is

Again by Lemma 2.7,

The contribution of this stratum in Theorem 6.2 is at most

Since $e\ge d$, this is at most

The ambient term contributes at most (ne). Hence

If $X$ is CPD-optimal, then $e=d$, and the bound becomes

The reverse inequality follows from Rouquier's lower bound and Lemma 2.6.

Corollary 8.2.

For every $d\ge0$ and $n\ge1$,

and

More generally, the same holds for $X[n]$ whenever $X$ is CPD-optimal.

9 Birational consequences in dimension at most three

In this section assume $k$ has characteristic zero.

Lemma 9.1.

Let

be a blow-up along a smooth center. If $Y$ is CPD-optimal, then $X$ is CPD-optimal.

Proof. By Lemma 5.4,

The reverse inequality follows from Rouquier's lower bound and Lemma 2.6. Hence $X$ is CPD-optimal.

Lemma 9.2.

If $C$ is a smooth proper curve, then

Proof. If $C\simeq\mathbb P^1$, then $C$ is CPD-optimal. If $C$ has positive genus, then its diagonal dimension is (2). Thus

and since

we get

Points have defect (0).

Theorem 9.3.

Over a field of characteristic zero, CPD-optimality is a birational invariant for smooth projective varieties of dimension at most three.

Proof. Let $X$ and $Y$ be smooth projective birational varieties of dimension at most three. By weak factorization, the birational map between them factors into a sequence of blow-ups and blow-downs along smooth centers.

A nontrivial smooth blow-up center in dimension at most three has dimension (0) or (1). Points have defect (0), and smooth proper curves have defect at most (1) by Lemma 9.2. Therefore Corollary 5.3 shows that blowing up preserves CPD-optimality, and Lemma 9.1 shows that blowing down preserves CPD-optimality.

Thus CPD-optimality is preserved along the factorization.

Corollary 9.4.

Every smooth projective rational variety of dimension at most three is CPD-optimal. In particular, if $X$ is such a variety, then

Proof. A smooth projective rational variety of dimension at most three is birational to $\mathbb P^d$. Projective space is CPD-optimal by Lemma 2.8. Apply Theorem 9.3 and Lemma 2.6.

10 Further directions

10.1 Equivariant and stacky versions

The arguments in this paper are non-equivariant. If a finite group $G$ acts on a building set $(X,\mathcal G)$, it is natural to ask for an equivariant version for quotient stacks

One expects a bound in terms of stabilizer-equivariant defects of the strata. Such a theorem would apply to unordered configuration compactifications and to quotient stacks such as

10.2 Castravet--Tevelev and Hassett spaces

The Castravet--Tevelev construction gives full $S_n$-invariant exceptional collections on $\overline M_{0,n}$ and passes through Hassett spaces. The proof in this paper bypasses this route for the non-equivariant Rouquier dimension. However, the equivariant problem should still require understanding the diagonal depth of the Hassett blocks appearing in their construction, especially the even/torsion blocks.

10.3 Beyond dimension three

The birational argument in Section 9 is special to dimension at most three. In dimension four, weak factorization may involve smooth surface centers. Smooth surfaces can have diagonal-depth defect larger than one, so Theorem 5.1 no longer implies birational invariance of optimality. The correct higher-dimensional birational statement should involve the maximal defect of centers appearing in a weak factorization.