The Degeneracy Loci for Smooth Moduli of Sheaves

Yu Zhao Beijing Institute of Technology [email protected]

Abstract.

Let $S$ be a smooth projective surface over $\mathbb{C}$ . We prove that, under certain technical assumptions, the degeneracy locus of the universal sheaf over the moduli space of stable sheaves is either empty or an irreducible Cohen-Macaulay variety of the expected dimension. We also provide a criterion for when the degeneracy locus is non-empty. This result generalizes the work of Bayer, Chen, and Jiang [5] for the Hilbert scheme of points on surfaces.

The above result is a special case of a general phenomenon: for a perfect complex of Tor-amplitude $[0,1]$ , the geometry of the degeneracy locus is closely related to the geometry of the derived Grassmannian. We analyze their birational geometry and relate it to the incidence varieties studied in [14]. As a corollary, we prove a statement that was previously claimed by the author in [37].

1. Introduction

1.1. The Degeneracy Loci for Smooth Moduli of Sheaves

Given a coherent sheaf $\mathcal{F}$ over a variety $X$ , the degeneracy locus $\bar{D}_{\mathcal{F},l}$ is defined as the closed subvariety

\bar{D}_{\mathcal{F},l}:=\{p\in X\mid\mathrm{rank}(\mathcal{F}|_{k(p)})\geq l\},\quad l\geq 0,

where $k(p)$ is the residue field of $p$ . The degeneracy locus plays a central role in Brill-Noether theory, enumerative geometry, and algebraic combinatorics, as discussed in [2, 10].

For the universal ideal sheaf over the Hilbert scheme of $n$ points on a projective surface $S$ over $\mathbb{C}$ , Chen, Jiang, and Bayer [5] proved that its degeneracy locus $\bar{D}_{\mathcal{F},l}$ is either empty or an irreducible Cohen-Macaulay variety of dimension

\mathrm{vdim}:=2n+2+l(1-l).

Moreover, they showed that $\bar{D}_{\mathcal{F},l}$ is non-empty if and only if $\mathrm{vdim}$ is positive.

In this paper, we generalize [5] to the rank $r>0$ moduli space of stable coherent sheaves. We fix $c_{1}\in\mathrm{NS}(S)$ and an ample line bundle $H$ . For any $n\in H^{4}(S,\mathbb{Z})\cong\mathbb{Z}$ , let $\mathcal{M}^{s}_{H}(r,c_{1},n)$ be the moduli space of Gieseker $H$ -stable coherent sheaves with the total Chern class

r+c_{1}+n[pts],

where $[pts]$ is the cycle of a point in $S$ . Let $\mathcal{U}_{n}$ be the universal coherent sheaf over $\mathcal{M}^{s}_{H}(r,c_{1},n)\times S$ . For $l\geq 0$ , the expected dimension of the degeneracy locus $\bar{D}_{\mathcal{U}_{n},l}$ is defined as

\mathrm{vdim}_{n,l}:=\dim(\mathcal{M}_{H}^{s}(r,c_{1},n))+\min\{0,l(r-l)\}.

We prove that

Theorem 1.1.

Assuming 3.8, for any $n\in\mathbb{Z}$ and $l\geq 0$ , the degeneracy locus $\bar{D}_{\mathcal{U}_{n},l}$ is either empty or an irreducible Cohen-Macaulay variety of dimension $\mathrm{vdim}_{n,l}$ .

Determining whether $\bar{D}_{\mathcal{U}_{n},l}$ is empty is more challenging than in the case of Hilbert schemes of points, due to the increased complexity of the moduli space and the involvement of the remainder of $l$ divided by $r$ . We prove that

Theorem 1.2.

There exists an integer $\theta\geq 0$ that depends on $S$ , $H$ , $r$ , and $c_{1}$ , such that $\bar{D}_{\mathcal{U}_{n},l}\neq\emptyset$ if and only if $\mathcal{M}_{H}^{s}(r,c_{1},n)\neq\emptyset$ and

\mathrm{vdim}_{n,l}\geq\theta+t(r-t),

where $t$ is the remainder of $l$ divided by $r$ .

Remark 1.3.

The precise value of $\theta$ is known in many cases. When the rank $r=1$ , we have $\theta=2$ , as shown in [5]. According to Yoshioka [33, 34], $\theta=2$ for K3 surfaces (for a generic ample divisor $H$ ) and $\theta=4$ for abelian surfaces. When $S=\mathbb{P}^{2}$ or $\mathbb{P}^{1}\times\mathbb{P}^{1}$ , the value of $\theta$ depends on $r$ and $c_{1}$ , as computed by Drezet and Le Potier [9].

Remark 1.4.

Similar arguments as in Theorem 1.1 and Theorem 1.2 also apply to $\mathcal{M}_{fr}(r,n)$ , the moduli space of framed sheaves on $\mathbb{P}^{2}$ , where we have $\theta=2$ . Since $\mathcal{M}_{fr}(r,n)$ is also the quiver variety of the Jordan quiver, we expect similar results for more quiver varieties, which will be studied in future work.

Remark 1.5.

Besides its importance in algebraic geometry, the enumerative and birational geometry properties of $\mathcal{U}_{n}$ play a central role in the representations of quantum toroidal algebras [27, 28], deformed $W$ -algebras [22], and categorification problems [36, 35]. These have further implications in the AGT correspondence [30] and the $P=W$ conjecture [12].

1.2. The Degeneracy Locus and the Grassmannian

One of the main ingredients in the proof of Theorem 1.1 follows from a duality relation between the degeneracy locus and the Grassmannian. Let $X$ be an irreducible, locally complete intersection variety over an arbitrary field $\mathbb{F}$ , and let

V\xrightarrow{\sigma}W

be a morphism of locally free sheaves over $X$ . Given any $l\geq 0$ , we consider the $l$ -th degeneracy locus $\bar{D}_{\sigma,l}:=\bar{D}_{\mathrm{coker}(\sigma),l}$ , i.e., the closed subvariety consisting of points where $\dim(\mathrm{coker}(\sigma))\leq l$ .

By a classical theorem in commutative algebra (e.g., Theorem 14.4 of [10]), for any $l\geq 0$ , we have

(1.1)

\mathrm{codim}(\bar{D}_{\sigma,l})\leq\max\{0,l(l-e)\},

where $e:=\mathrm{rank}(W)-\mathrm{rank}(V)$ . We say that $\bar{D}_{\sigma,l}$ is of expected dimension if equality holds in 1.1; in this situation, $\bar{D}_{\sigma,l}$ is Cohen-Macaulay.

On the other hand, for any integer $l\geq 0$ , we consider the Grassmannian (in Grothendieck’s notation) $Gr_{X}(\mathrm{coker}(\sigma),l)$ , which parametrizes the quotients of $\sigma$ by locally free sheaves of rank $l$ . By Krull’s height theorem,

(1.2)

\dim(Gr_{X}(\mathrm{coker}(\sigma),l))\geq\dim(X)-l(l-e),

and we say that $Gr_{X}(\mathrm{coker}(\sigma),l)$ is of expected dimension if equality holds in 1.2 or if $Gr_{X}(\mathrm{coker}(\sigma),l)=\emptyset$ . Theorem 1.1 follows from Nakajima-Yoshioka’s theory on perverse coherent sheaves of blow-ups [25, 24, 26] and the following proposition.

Proposition 1.6.

The following are equivalent:

(1)

For all $l\geq 0$ , the $l$ -th degeneracy locus $\bar{D}_{\sigma,l}$ is of expected dimension.
(2)

For all $l\geq 0$ , the $l$ -th degeneracy locus $\bar{D}_{\sigma^{\vee},l}$ is of expected dimension.
(3)

For all $l\geq 0$ , the rank $l$ Grassmannians $Gr_{X}(\mathrm{coker}(\sigma),l)$ are of expected dimension.
(4)

For all $l\geq 0$ , the rank $l$ Grassmannians $Gr_{X}(\mathrm{coker}(\sigma^{\vee}),l)$ are of expected dimension.

Moreover, if one of these four families of varieties consists entirely of irreducible varieties, then the other three families are also irreducible.

Remark 1.7.

Under the assumptions of Proposition 1.6, more homological and birational properties of the degeneracy loci and Grassmannians can be studied. For more details, we refer to [1, Appendix A].

Remark 1.8.

While preparing this paper, the author was reminded by Qingyuan Jiang that a similar argument to Proposition 1.6 was given in his note [17].

Following Proposition 1.6, we give the following definition:

Definition 1.9.

We say that $\sigma$ has good degeneracy loci if the assumptions in Proposition 1.6 are satisfied. Moreover, we say $\sigma$ has perfect degeneracy loci if all the varieties in Proposition 1.6 are irreducible.

1.3. Incidence Varieties, Birational Geometry, and Derived Algebraic Geometry

To prove Theorem 1.2, we need to consider incidence varieties and study their birational geometry, adopting a higher viewpoint from derived algebraic geometry.

First, we observe that the morphism $\sigma$ between two locally free sheaves can be naturally generalized to a Tor-amplitude $[0,1]$ perfect complex. That is, we assume $\sigma$ is a two-term complex of locally free sheaves, locally. Let $e$ denote the rank of $\sigma$ . Then, for any point $p\in X$ , we have the Fredholm relation

(1.3)

\dim\ h^{0}(\sigma|_{k(p)})-\dim\ h^{1}(\sigma|_{k(p)})=e,

where $k(p)$ is the residue field at $p$ .

For any $l\geq 0$ , Jiang [14] constructed a derived Grassmannian $Gr_{X}(\sigma,l)$ , which is a derived enhancement of the Grassmannian

Gr_{X}(h^{0}(\sigma),l).

In particular, Jiang [14] proved that $Gr_{X}(\sigma,l)$ is quasi-smooth, which generalizes the notion of locally complete intersection varieties in derived algebraic geometry. If $\sigma$ is a Tor-amplitude $[0,1]$ perfect complex, then so is the shifted derived dual $\sigma^{\vee}[1]$ . When $e:=\mathrm{rank}(\sigma)\geq 0$ , for any two integers $d_{+},d_{-}\geq 0$ , Jiang [14] defined an incidence variety

\mathrm{Inc}_{X}(\sigma,d_{+},d_{-}),

which is a fiber product of $Gr_{X}(\sigma,d_{+})$ and $Gr_{X}(\sigma^{\vee}[1],d_{-})$ over $X$ , albeit with different derived structures. Specifically, we have

\mathrm{Inc}_{X}(\sigma,d_{+},0)\cong Gr_{X}(\sigma,d_{+}),\quad\mathrm{Inc}_{X}(\sigma,0,d_{-})\cong Gr_{X}(\sigma^{\vee}[1],d_{-}).

Given a quasi-smooth derived scheme $Y$ over $\mathbb{F}$ , we say that $Y$ is classical if it has a trivial derived structure, i.e., $\pi_{0}(Y)\cong Y$ , where $\pi_{0}(Y)$ is the underlying scheme. By Corollary 2.11 of [3], $Y$ is classical if and only if it has the expected dimension, i.e.,

\mathrm{vdim}(Y)=\dim(\pi_{0}(Y)),

where $\mathrm{vdim}(Y)$ is the virtual dimension. Moreover, $Y$ is always a locally complete intersection variety if $Y$ is classical. Hence, Proposition 1.6 can be derived from the following generalization:

Theorem 1.10.

Assuming that $e:=\mathrm{rank}(\sigma)\geq 0$ , the following are equivalent:

(1)

$\sigma$ has good (resp. perfect) degeneracy loci;
(2)

For any integers $d_{+},d_{-}\geq 0$ , the incidence variety

$\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})$

is classical (resp. and irreducible);
(3)

For any integer $d_{+}\geq 0$ , the derived Grassmannian

$Gr_{X}(\sigma,d_{+})$

is classical (resp. and irreducible);
(4)

For any integer $d_{-}\geq 0$ , the derived Grassmannian

$Gr_{X}(\sigma^{\vee}[1],d_{-})$

is classical (resp. and irreducible).

When the assumptions of Theorem 1.10 are satisfied, the derived Grassmannians and incidence varieties exhibit the following birational property:

Proposition 1.11 (Proposition 2.17).

Let $\sigma$ be a Tor-amplitude $[0,1]$ complex of rank $e\geq 0$ with perfect degeneracy loci. Then for $d_{+},d_{-}\geq 0$ such that $d_{+}=d_{-}+e$ , the morphisms

	$\displaystyle r^{+}:\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})\to Gr_{X}(\sigma,d_{+}),\quad$	$\displaystyle r^{-}:\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})\to Gr_{X}(\sigma^{\vee}[1],d_{-}),$
	$\displaystyle\mathrm{pr}_{\sigma,d_{+}}:Gr_{X}(\sigma,d_{+})\to\bar{D}_{\sigma,d_{+}},\quad$	$\displaystyle\mathrm{pr}_{\sigma,d_{-}}:Gr_{X}(\sigma^{\vee}[1],d_{-})\to\bar{D}_{\sigma,d_{+}}$

are birational. Moreover, all the above morphisms are isomorphisms if

d_{+}d_{-}\leq\dim(X)<(d_{+}+1)(d_{-}+1).

By Proposition 1.11, when all $Gr_{X}(\sigma,d_{+})$ and $Gr_{X}(\sigma^{\vee}[1],d_{-})$ are smooth varieties of expected dimensions, the main theorem of [32, 16] provides new examples of the DK conjecture of Bondal-Orlov [7] and Kawamata [18].

1.4. The Virtual Fundamental Class of Incidence Varieties

Let $\sigma$ be a Tor-amplitude $[0,1]$ perfect complex over a proper smooth $\mathbb{C}$ -variety $X$ such that $\mathrm{rank}(\sigma)\geq 0$ . There are two natural projection morphisms (given by the fiber diagram):

r_{+}:\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})\to Gr_{X}(\sigma,d_{+}),\quad r_{-}:\mathrm{Inc}_{X}(\sigma^{\vee}[1],d_{-})\to Gr_{X}(\sigma^{\vee}[1],d_{-}).

According to [14], the morphisms $r_{+}$ and $r_{-}$ are quasi-smooth, and $\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})$ is also quasi-smooth. Following [8], this quasi-smooth structure induces a perfect obstruction theory in the sense of Li-Tian [21] or Behrend-Fantechi [6], thereby inducing a virtual fundamental class

[\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})]^{\mathrm{vir}}\in CH_{d_{0}}(\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})),

where

d_{0}=\dim(X)+(1-d_{+})d_{+}-(d_{-}+1)d_{-}+d_{-}d_{+}.

In particular, if all $Gr_{X}(\sigma,d_{+})$ and $Gr_{X}(\sigma^{\vee}[1],d_{-})$ are smooth varieties of expected dimensions, the virtual fundamental class $[\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})]^{\mathrm{vir}}$ can be viewed as a correspondence in

CH^{*}(Gr_{X}(\sigma,d_{+})\times Gr_{X}(\sigma^{\vee}[1],d_{-})).

This correspondence plays a crucial role in the author’s earlier paper [37], which studies the Clifford algebra action on the cohomology of the moduli space of rank $1$ perverse coherent sheaves.

Now, by Theorem 1.10, all the incidence varieties $\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})$ are locally complete intersection varieties of expected dimensions, thereby proving an earlier argument of the author in [37].

Proposition 1.12 (Proposition 4.6 of [37]).

We have

[\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})]^{\mathrm{vir}}=[\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})]\in CH_{d_{0}}(\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})).

1.5. Perverse Coherent Sheaves on Blow-ups and Derived Grassmannians

For the tautological bundles/sheaves over moduli spaces, the geometry of derived Grassmannians can often be understood more easily than that of degeneracy loci, as the derived Grassmannians themselves can be described as moduli spaces. A notable example stems from the ADHM construction, which classifies anti-self-dual instantons over $\mathbb{R}^{4}$ . Given integers $r>0$ and $n\geq 0$ , we consider two vector spaces $V$ and $W$ with dimensions $n$ and $r$ , respectively. The ADHM data is defined as

M^{s}(V,W):=\{(X,Y,i,j)\in\mathrm{Hom}(V,V)\oplus\mathrm{Hom}(V,V)\oplus\mathrm{Hom}(W,V)\oplus\mathrm{Hom}(V,W)\},

subject to the equation

[X,Y]+ij=0,

and the stability condition that for any subspace $V^{\prime}\subset V$ , closed under $X$ and $Y$ and contained in the kernel of $j$ , we must have $V^{\prime}=0$ . The moduli space of anti-self-dual instantons is given as the geometric quotient

\mathcal{M}_{fr}(r,n):=M^{s}(V,W)/\mathrm{GL}(V),

which is a smooth variety of dimension $2rn$ . The variety $\mathcal{M}_{\mathrm{fr}}(r,n)$ also coincides with the moduli space of framed sheaves over $\mathbb{P}^{2}$ and the quiver variety for the Jordan quiver (i.e., a point with one loop).

The universal sheaf $\mathcal{U}_{0}$ on $\mathcal{M}_{fr}(r,n)$ is represented by the complex

\mathcal{V}\xrightarrow{X\oplus Y\oplus j}\mathcal{V}\oplus\mathcal{V}\oplus\mathcal{W}\xrightarrow{Y\oplus-X\oplus i}\mathcal{V},

which is a two-term complex, as $X\oplus Y\oplus j$ is injective at any closed point by the stability condition. The derived Grassmannians of $\mathcal{U}_{0}$ and $\mathcal{U}_{0}^{\vee}[1]$ were described by Nakajima and Yoshioka [25] via a quiver description. They further generalized this to the moduli space of stable sheaves over a surface, which are known as perverse coherent sheaves on blow-ups [24]. This framework forms another essential component of our proof for Theorem 1.1 and Theorem 1.2.

Recently, the geometric representation theory of the moduli space of perverse coherent sheaves has gained attention, particularly because their cohomology groups form a representation of the affine super-Yangian of $\mathfrak{gl}(1|1)$ . For further details, we refer to [29, 11, 37].

1.6. Some Notations

The results in Section 2 hold for an arbitrary field $\mathbb{F}$ . In Section 3, we will focus specifically on the field of complex numbers $\mathbb{C}$ .

1.7. Acknowledgement

I would like to thank Qingyuan Jiang for many helpful discussions.

2. Degeneracy Theory for Tor-Amplitude $[0,1]$ -Complexes

In this section, we assume that $X$ is a locally complete intersection variety over a field $\mathbb{F}$ , and $\sigma$ is a rank $e$ Tor-amplitude $[0,1]$ -complex.

2.1. Degeneracy Loci of a Tor-Amplitude $[0,1]$ -Complex

Definition 2.1 (Section 14.4 of [10]).

For any integer $l$ , we define the $l$ -th degeneracy locus as

\bar{D}_{\sigma,l}:=\{x\mid\dim\ h^{0}(\sigma|_{k(x)})\geq l\},

which is a closed subvariety of $X$ , where $k(x)$ denotes the residue field of $x$ . We also define the locally closed subvariety

D_{\sigma,l}:=\bar{D}_{\sigma,l}-\bar{D}_{\sigma,l+1}=\{x\mid\dim\ h^{0}(\sigma|_{k(x)})=l\}.

By the definition of degeneracy loci, we have:

\bar{D}_{\sigma,k}\cong X\quad\mathrm{if}\quad k\leq\min\{0,e\},\quad D_{\sigma,k}\cong\emptyset\quad\mathrm{if}\quad k<\min\{0,e\},

and the duality relation:

\bar{D}_{\sigma,k}=\bar{D}_{\sigma^{\vee}[1],k-e}.

Example 2.2.

Given two integers $m,n>0$ , consider the affine variety

\mathbb{V}_{m,n}:=|\mathrm{Hom}(\mathbb{F}^{m},\mathbb{F}^{n})|.

There is a universal map of locally free sheaves over $\mathbb{V}_{m,n}$ :

v_{m,n}:\mathcal{O}^{m}\to\mathcal{O}^{n},

which can be regarded as a Tor-amplitude $[0,1]$ -complex. The degeneracy loci are given by:

\bar{D}_{v_{m,n},k}=\{E\mid\mathrm{rank}(E)\leq n-k\},\quad D_{v_{m,n},k}=\{E\mid\mathrm{rank}(E)=n-k\},

which are Cohen-Macaulay and locally complete intersection varieties of codimension

\max\{0,k(k+m-n)\}.

In general, we have the following property for the Cohen-Macaulay loci:

Proposition 2.3 (Theorem 14.4 (b)(c) of [10]).

For any integer $k$ , the codimension of the degeneracy loci satisfies the inequalities:

\mathrm{codim}(\bar{D}_{\sigma,k})\leq\max\{0,k(k-e)\},\quad\mathrm{codim}(D_{\sigma,k})\leq\max\{0,k(k-e)\}.

In particular, if equality holds, then $\bar{D}_{\sigma,k}$ (resp. $D_{\sigma,k}$ ) is Cohen-Macaulay (resp. a locally complete intersection variety).

Proof.

As the argument is local, we can assume that $X$ is affine and

\sigma\cong\{\mathcal{O}^{m}\xrightarrow{s}\mathcal{O}^{n}\}.

Then the argument follows directly from Theorem 14.4 (b)(c) of [10]. ∎

Definition 2.4.

We say that $\sigma$ has good degeneracy loci if for any $k\geq 0$ ,

\mathrm{codim}(\bar{D}_{\sigma,k})=\max\{0,k(k-e)\}.

If $\sigma$ has good degeneracy loci, we call the degeneracy loci perfect if $\bar{D}_{\sigma,k}$ is irreducible for all $k\geq 0$ .

The following properties of the degeneracy loci follow from Proposition 2.3 and the Fredholm relation 1.3.

Lemma 2.5.

The complex $\sigma$ has good (resp. perfect) degeneracy loci if and only if $\sigma^{\vee}[1]$ has good (resp. perfect) degeneracy loci.

Lemma 2.6.

If $\sigma$ has good degeneracy loci, then any degeneracy locus $\bar{D}_{\sigma,k}$ is Cohen-Macaulay, and $D_{\sigma,k}$ is a locally complete intersection variety.

Lemma 2.7.

The complex $\sigma$ has good (resp. perfect) degeneracy loci if and only if for any $k\geq 0$ ,

\mathrm{codim}(D_{\sigma,k})=\max\{0,k(k-e)\},

(and $D_{\sigma,k}$ is irreducible).

2.2. Derived Grassmannians of a Tor-Amplitude $[0,1]$ -Complex

Definition 2.8 (Definition 4.3 of [14]).

Given an integer $d\geq 0$ , the derived Grassmannian $Gr_{X}(\sigma,d)$ is defined as the moduli space of quotients

\{\sigma\twoheadrightarrow\mathcal{V}\},

where $\mathcal{V}$ is a rank $d$ locally free sheaf. We define

pr_{\sigma,d}:Gr_{X}(\sigma,d)\to X

as the projection morphism.

Remark 2.9.

It is notable that we always have $Gr_{X}(\sigma,0)\cong X$ .

The notion of quasi-smooth morphisms and schemes generalizes locally complete intersection morphisms and schemes in derived algebraic geometry:

Definition 2.10.

A morphism of derived schemes $f:X\to Y$ is defined to be quasi-smooth if the cotangent complex $L_{f}:=L_{X/Y}$ has Tor-amplitude $[0,1]$ . A derived scheme $X$ is defined to be quasi-smooth if the morphism from $X$ to $\mathrm{Spec}(\mathbb{C})$ is quasi-smooth. For a quasi-smooth morphism $f$ (resp. scheme $X$ ), we define its virtual dimension, denoted

vdim(f),\quad\text{(resp. }vdim(X)),

as the rank of the cotangent complex $L_{f}$ (resp. $L_{X}$ ). The relative canonical bundle $K_{f}$ (resp. $K_{X}$ ) is defined as the determinant of $L_{f}$ (resp. $L_{X}$ ). (We refer to [31] or [15] for details on the determinant of a Tor-amplitude $[0,1]$ complex.)

Theorem 2.11 ([14]).

The derived Grassmannian $Gr_{X}(\sigma,d)$ is a proper derived scheme over $X$ , and the projection map

pr_{\sigma,d}:Gr_{X}(\sigma,d)\to X

is quasi-smooth, with relative virtual dimension $d(e-d)$ .

We observe that if $\sigma$ has Tor-amplitude $[0,1]$ , then the shifted dual $\sigma^{\vee}[1]$ also has Tor-amplitude $[0,1]$ .

Definition 2.12 (Incidence Correspondence, Definition 2.7 of [16]).

Given $(d_{+},d_{-})\in\mathbb{Z}_{\geq 0}^{2}$ , the incidence scheme $\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})$ can be defined in two ways, which were proven equivalent in [14]:

(1)

Over the projection morphism $pr_{+}:Gr_{X}(\sigma,d_{+})\to X$ , let $\sigma_{+}$ be the fiber of the universal quotient:

$\theta_{+}:pr^{*}_{+}\sigma\to\mathcal{V}^{+}.$

We define

$\mathrm{Inc}_{X}(\sigma,d_{+},d_{-}):=Gr_{Gr_{X}(\sigma,d_{+})}(\sigma_{+}^{\vee}[1],d_{-}).$
(2)

Over the projection morphism $pr_{-}:Gr_{X}(\sigma^{\vee}[1],d_{-})\to X$ , let $\sigma_{-}$ be the fiber of the universal quotient:

$\theta_{-}:pr^{*}_{-}\sigma^{\vee}[1]\to\mathcal{V}^{-\vee}.$

We define

(2.1) $\mathrm{Inc}_{X}(\sigma,d_{+},d_{-}):=Gr_{Gr_{X}(\sigma^{\vee}[1],d_{-})}(\sigma_{-}^{\vee}[1],d_{+}).$

The above definitions induce canonical projection morphisms:

r^{+}_{d_{+},d_{-}}:\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})\to Gr_{X}(\sigma,d_{+}),\quad r^{-}_{d_{+},d_{-}}:\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})\to Gr_{X}(\sigma^{\vee}[-1],d_{-}),

which we will abbreviate as $r^{+}$ and $r^{-}$ respectively when context allows.

2.3. Degeneracy Loci and the Classicality of Incidence Varieties

In this subsection, we show the relation between the classicality of incidence varieties and the goodness of the degeneracy loci:

Theorem 2.13.

Assuming that $e:=\mathrm{rank}(\sigma)\geq 0$ , the following are equivalent:

(1)

$\sigma$ has a good (resp. perfect) degeneracy loci.
(2)

For any integer $d_{+},d_{-}\geq 0$ , the incidence variety

$\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})$

is classical (resp. and irreducible).
(3)

For any integer $d_{+}\geq 0$ , the derived Grassmannian

$Gr_{X}(\sigma,d_{+})$

is classical (resp. and irreducible).
(4)

For any integer $d_{-}\geq 0$ , the derived Grassmannian

$Gr_{X}(\sigma^{\vee}[1],d_{-})$

is classical (resp. and irreducible).

Theorem 2.13 follows from Lemma 2.5 and the following two propositions:

Proposition 2.14.

Assuming $e:=\mathrm{rank}(\sigma)\geq 0$ , if $\sigma$ has a good (resp. perfect) degeneracy loci, the incidence variety

\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})

is classical (resp. and irreducible).

Proposition 2.15.

If for any $d\geq 0$ , the derived Grassmannian

Gr_{X}(\sigma,d)

is classical (resp. and irreducible), then $\sigma$ has good (resp. perfect) degeneracy loci.

The key point of Propositions 2.14 and 2.15 is the following lemma:

Lemma 2.16.

Over $D_{\sigma,l}$ , the underlying map of the projection from $\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})$ to $X$ is a $Gr(l,d_{+})\times Gr(l-e,d_{-})$ bundle.

Proof.

This directly follows from the definition of $D_{\sigma,l}$ . ∎

Proof of Proposition 2.14.

Let $\sigma$ be a good degeneracy loci of rank $e\geq 0$ and $t$ be the projection from the incidence variety to $X$ . Then the underlying scheme of the preimage over $D_{\sigma,l}$ is empty if $l<\max\{d_{+},e+d_{-}\}$ and has dimension

	$\displaystyle\dim(X)-\mathrm{codim}(D_{\sigma,l})+d_{+}(l-d_{+})+d_{-}(l-e-d_{-})$
	$\displaystyle=\dim(X)-l(l-e)+d_{+}(l-d_{+})+d_{-}(l-e-d_{-})$
	$\displaystyle=\mathrm{vdim}(\mathrm{Inc}_{X}(\sigma,d_{+},d_{-}))-(l-d_{+})(l-d_{-}-e)\leq\mathrm{vdim}(\mathrm{Inc}_{X}(\sigma,d_{+},d_{-}))$

if $l\geq\max\{d_{+},e+d_{-}\}$ . Moreover, equality holds if and only if $l=\max\{d_{+},e+d_{-}\}$ . Hence, if $\sigma$ has good degeneracy loci, we have

\dim(\pi_{0}(\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})))=\mathrm{vdim}(\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})),

and thus it is classical. Moreover, if $\sigma$ has perfect degeneracy loci, $\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})$ has only one irreducible component of maximal dimension and is therefore also irreducible. ∎

Proof of Proposition 2.15.

For any $d\geq\max\{0,e\}$ , the underlying scheme map of $pr_{\sigma,d}$ over $D_{\sigma,d}$ is an isomorphism. Thus, if the underlying scheme of the derived Grassmannian

Gr_{X}(\sigma,d)

is irreducible, so is $D_{\sigma,d}$ . Moreover, if the derived Grassmannian is classical, then

\dim(X)-\mathrm{codim}(D_{\sigma,d})=\dim(D_{\sigma,d})\leq\mathrm{vdim}(Gr_{X}(\sigma,d))=\dim(X)-d(d-e),

and hence $\mathrm{codim}(D_{\sigma,d})=d(d-e)$ by Proposition 2.3. Thus, by Lemma 2.5, $\sigma$ has a good degeneracy loci and a perfect degeneracy loci if all the derived Grassmannians are irreducible. ∎

2.4. Birational Geometry of Incidence Varieties for Perfect Degeneracy Loci

In this subsection, we consider the birational geometry of incidence varieties. By Lemma 2.16, we have the following result:

Proposition 2.17 (Proposition 1.11).

Let $\sigma$ be a tor-amplitude $[0,1]$ complex of rank $e\geq 0$ with perfect degeneracy loci. Then, for integers $d_{+},d_{-}\geq 0$ such that $d_{+}=d_{-}+e$ , the morphisms

	$\displaystyle r^{+}:\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})$	$\displaystyle\to Gr_{X}(\sigma,d_{+}),$
	$\displaystyle r^{-}:\mathrm{Inc}_{X}(\sigma,d_{+},d_{-})$	$\displaystyle\to Gr_{X}(\sigma^{\vee}[1],d_{-}),$
	$\displaystyle pr_{\sigma,d_{+}}:Gr_{X}(\sigma,d_{+})$	$\displaystyle\to\bar{D}_{\sigma,d_{+}},$
	$\displaystyle pr_{\sigma,d_{-}}:Gr_{X}(\sigma^{\vee}[1],d_{-})$	$\displaystyle\to\bar{D}_{\sigma,d_{+}}$

are birational. Furthermore, all of the above morphisms are isomorphisms if

d_{+}d_{-}\leq\dim(X)<(d_{+}+1)(d_{-}+1).

Proof.

We only need to observe that the projections

pr_{\sigma,d_{+}},\quad pr_{\sigma^{\vee}[1],d_{-}}

are isomorphisms over $D_{\sigma,d_{+}}$ . Moreover, when

d_{+}d_{-}\leq\dim(X)<(d_{+}+1)(d_{-}+1),

the degeneracy locus $\bar{D}_{\sigma,d_{+}+1}$ is empty, since its dimension is negative. ∎

3. Degeneracy Loci on the Smooth Moduli Surface of Coherent Sheaves over a Surface

Let $S$ be a smooth projective surface over $\mathbb{C}$ . Fix an ample line bundle $H$ , an integer $r\in\mathbb{Z}_{>0}$ , and a class $c_{1}\in H^{2}(S,\mathbb{Z})$ . Given $n\in H^{4}(S,\mathbb{Z})\cong\mathbb{Z}$ , we denote by

\mathcal{M}_{H}^{ss}(r,c_{1},n)\quad\text{and}\quad\mathcal{M}_{H}^{s}(r,c_{1},n)

the moduli spaces of Gieseker $H$ -semistable (resp. stable) sheaves on $S$ with Chern classes $(r,c_{1},n)$ . Let

\mathcal{U}\in\mathrm{Coh}(\mathcal{M}_{H}^{s}(r,c_{1},n)\times S)

be the universal coherent sheaf.

In this section, we prove Theorem 1.1 and Theorem 1.2.

3.1. A General Proposition

We first state two general propositions:

Proposition 3.1.

Let $\{X_{m}\}_{m\in\mathbb{Z}}$ be a family of irreducible smooth varieties over $\mathbb{C}$ with constant integers const and $r>0$ such that for each $m\in\mathbb{Z}$ , $X_{m}$ is either empty or an irreducible smooth variety of dimension

const+2rm.

Let $\mathcal{V}:=\{\mathcal{V}_{m}\in\text{Perf}(X_{m})\}$ be a family of tor-amplitude $[0,1]$ and fixed rank $r$ complexes such that for all $m,l$ ,

Gr_{X_{m+l}}(\mathcal{V}^{\vee}_{m+l}[1],l)\cong Gr_{X_{m}}(\mathcal{V}_{m},l)

are classical irreducible varieties. Then the complex $\mathcal{V}$ has perfect degeneracy loci for any $X_{m}$ .

Proof.

This follows from Theorem 2.13. ∎

Proposition 3.2.

Assuming the setting of Proposition 3.1, if there exists an integer $\theta_{0}$ such that $X_{m}$ is empty if and only if

const+2rm\geq\theta_{0},

then the degeneracy locus

\bar{D}_{\mathcal{V}_{m},l}\neq\emptyset

if and only if the expected dimension

const+2rm+l(r-l)-t(r-t)\geq\theta_{0},

where $t$ is the remainder of $l$ divided by $r$ .

Proof.

We make induction on $l$ . When $l<r$ , the left side of the inequality is the dimension of $X_{m}$ , and $\bar{D}_{\mathcal{V}_{m},l}$ is not empty if and only if $X_{m}$ is not empty. Thus, the statement holds in this case.

When $l\geq r$ , we have

\bar{D}_{\mathcal{V}_{m},l}=\bar{D}_{\mathcal{V}_{m}^{\vee}[1],l-r},

which is birational to $Gr_{X_{m-l+r}}(\mathcal{V},l-r)$ . Therefore, $\bar{D}_{\mathcal{V}_{m},l}$ is not empty if and only if $\bar{D}_{\mathcal{V}_{m-l+r},l-r}$ is not empty. By induction, this holds if and only if

const+2r(m-l+r)+(l-r)(2r-l)-t(r-t)\geq\theta_{0}.

Simplifying the right-hand side, we get

const+2rm+l(r-l)-t(r-t)\geq\theta_{0}.

∎

Corollary 3.3.

Let

vdim_{m,l}:=\dim(X_{m})+\max\{0,l(r-l)\}.

Then the degeneracy locus

\bar{D}_{\mathcal{V}_{m},l}\neq\emptyset

if and only if $X_{m}\neq\emptyset$ and

vdim_{m,l}\geq\theta_{0}+t(r-t),

where $t$ is the remainder of $l$ divided by $r$ .

3.2. Moduli Space of Framed Coherent Sheaves on the Plane and ADHM Construction

Given integers $(r,n)\in\mathbb{Z}_{>0}\times\mathbb{Z}_{\geq 0}$ , we denote by

\mathcal{M}_{fr}(r,n)

the moduli space of framed sheaves on the projective plane. This moduli space can also be interpreted as the quiver variety of the Jordan quiver. Specifically, we consider two vector spaces $V$ and $W$ of dimensions $n$ and $r$ , respectively. The ADHM datum for this setup is given by

M^{s}(V,W):=\{(X,Y,i,j)\in\operatorname{Hom}(V,V)\oplus\operatorname{Hom}(V,V)\oplus\operatorname{Hom}(W,V)\oplus\operatorname{Hom}(V,W)\}

subject to the equation

[X,Y]+ij=0,

along with the stability condition that for any subspace $V^{\prime}\subset V$ closed under $X$ and $Y$ and in the kernel of $j$ , we must have $V^{\prime}=0$ .

Theorem 3.4 (Chapter 2 of [23]).

\mathcal{M}_{fr}(r,n)\cong M^{s}(V,W)/GL(V),

where the general linear group $GL(V)$ acts by conjugation. Moreover, $\mathcal{M}_{fr}(r,n)$ is a non-empty smooth variety of dimension $2rn$ .

The vector spaces $V$ and $W$ induce tautological locally free sheaves, denoted by $\mathcal{V}$ and $\mathcal{W}$ . The universal sheaf $\mathcal{U}$ on $\mathcal{M}_{fr}(r,n)\times\mathbb{A}^{2}$ is represented by the two-term complex

\mathcal{V}\xrightarrow{(X-x\cdot\text{id})\oplus(Y-y\cdot\text{id})\oplus j}\mathcal{V}\oplus\mathcal{V}\oplus\mathcal{W}\xrightarrow{(Y-y\cdot\text{id})\oplus-(X-x\cdot\text{id})\oplus i}\mathcal{V},

for $(x,y)\in\mathbb{A}^{2}$ . This is a two-term complex because the map

(X-x\cdot\text{id})\oplus(Y-y\cdot\text{id})\oplus j

is injective at any closed point by the stability condition. We denote by $\mathcal{U}_{0}$ the restriction of $\mathcal{U}$ to $\mathcal{M}_{fr}(r,n)\times\{(0,0)\}$ , i.e., setting $x=y=0$ .

The derived Grassmannian of $\mathcal{U}$ is described by the perverse coherent sheaves of Nakajima-Yoshioka [25], which also admit a quiver-like description. Let $V_{1}$ , $V_{2}$ , and $W$ be vector spaces of dimensions $n_{1}$ , $n_{2}$ , and $r$ , respectively. The datum $M^{s}(V_{1},V_{2},W)$ includes elements $(B_{1},B_{2},d,i,j)$ in

\operatorname{Hom}(V_{1},V_{2})\oplus\operatorname{Hom}(V_{2},V_{1})\oplus\operatorname{Hom}(W,V_{1})\oplus\operatorname{Hom}(V_{2},W),

satisfying the equation

B_{1}dB_{2}-B_{2}dB_{1}=0,

and the stability condition that for any subspaces

V_{1}^{\prime}\subset V_{1},\quad V_{2}^{\prime}\subset V_{2},\quad 0\subset W

which are closed under the quiver representation, we must have $V_{1}^{\prime}=0$ . By the stability condition, $d$ is injective, implying $n_{1}\geq n_{2}$ .

Theorem 3.5 (Nakajima-Yoshioka [25]).

The moduli space

\mathcal{M}_{\text{perv}}(n_{1},n_{2},r):=M^{s}(V_{1},V_{2},W)/(GL(V_{1})\times GL(V_{2}))

is either an irreducible smooth variety of dimension $2n_{1}n_{2}-n_{1}^{2}-n_{2}^{2}+(n_{1}+n_{2})r$ or an empty set.

We define the morphisms

	$\displaystyle\zeta:\mathcal{M}_{\text{perv}}(n_{1},n_{2},r)\to\mathcal{M}(n_{1},r),$
	$\displaystyle(B_{1},B_{2},d,i,j)\mapsto(dB_{1},dB_{2},i,dj),$

and

	$\displaystyle\eta:\mathcal{M}_{\text{perv}}(n_{1},n_{2},r)\to\mathcal{M}(n_{2},r),$
	$\displaystyle(B_{1},B_{2},d,i,j)\mapsto(B_{1}d,B_{2}d,id,j),$

which are well-defined by [25].

Theorem 3.6 ([25]).

We have

	$\displaystyle\mathcal{M}_{\text{perv}}(n_{1},n_{2},r)$	$\displaystyle\cong\operatorname{Gr}_{\mathcal{M}_{fr}(n_{1},r)}(\mathcal{U}_{0}^{\vee}[1],n_{1}-n_{2}),$
	$\displaystyle\mathcal{M}_{\text{perv}}(n_{1},n_{2},r)$	$\displaystyle\cong\operatorname{Gr}_{\mathcal{M}_{fr}(n_{2},r)}(\mathcal{U}_{0},n_{1}-n_{2}),$

where $\zeta$ and $\eta$ are the projection morphisms.

Applying Theorem 3.6, we conclude:

Corollary 3.7.

Fixing $r>0$ , the sheaf $\mathcal{U}_{0}\in\coprod_{n}\operatorname{Coh}(\mathcal{M}_{fr}(r,n))$ satisfies the assumptions in Proposition 3.1, with

const=\theta_{0}=0.

In particular, $\mathcal{U}_{0}\in\coprod_{n}\operatorname{Coh}(\mathcal{M}_{fr}(r,n))$ and $\mathcal{U}\in\coprod_{n}\operatorname{Coh}(\mathcal{M}_{fr}(r,n))\times\mathbb{A}^{2}$ both have perfect degeneracy loci. The degeneracy locus

\bar{D}_{\mathcal{U}_{0},l}\quad\text{and}\quad\bar{D}_{\mathcal{U},l}

is non-empty if and only if

2rn+l(r-l)-t(r-t)\geq 0,

where $t$ is the remainder when $l$ is divided by $r$ .

3.3. Moduli Space of Gieseker Stable Sheaves

The geometry of the moduli space of Gieseker stable coherent sheaves can be complex. To simplify computations, we assume the following conditions:

Assumption 3.8.

We make the following assumptions:

(1)

The total Chern class is primitive, i.e.,

$\gcd(r,c_{1}\cdot H)=1.$
(2)

Either $r=1$ , or $K_{S}\cong\mathcal{O}_{S}$ , or $c_{1}(K_{S})\cdot H<0$ .
(3)

There exists a semistable coherent sheaf $\mathcal{F}$ of rank $r$ with first Chern character $c_{1}$ , i.e.,

$\coprod_{n\in\mathbb{Z}}\mathcal{M}^{ss}_{H}(r,c_{1},n)\neq\emptyset.$

Remark 3.9.

3.8 holds for generic ample divisors when the surface $S$ is a Del Pezzo, K3, or abelian surface.

We state the following general results concerning the moduli space $\mathcal{M}_{H}^{s}(r,c_{1},n)$ :

Proposition 3.10 ([13]; see also Proposition 2.10 of [28] and Lemma 3.3 of [19]).

If 3.8 holds, then

\mathcal{M}_{H}^{ss}(r,c_{1},n)=\mathcal{M}_{H}^{s}(r,c_{1},n),

which is either empty or a smooth projective variety of dimension

\dim\mathcal{M}_{H}^{s}(r,c_{1},n)=const+2rn,

where

const:=(1-r)c_{1}^{2}-(r^{2}-1)\chi(\mathcal{O}_{S})+h^{1}(\mathcal{O}_{S}).

Proposition 3.11 (Proposition 2.14 of [28]).

Assuming 3.8, the universal sheaf $\mathcal{U}$ , regarded as an object in $\mathrm{Perf}(\mathcal{M}_{H}^{s}(r,c_{1},n)\times S)$ , has tor-amplitude in $[0,1]$ .

Lemma 3.12 (Proposition 5.5 of [27]).

Under 3.8, for any non-splitting short exact sequence

0\to\mathcal{U}^{\prime}\to\mathcal{U}\to\mathbb{C}_{x}\to 0,

where $x\in S$ and both $\mathcal{U}^{\prime}$ and $\mathcal{U}$ have rank $r$ and first Chern character $c_{1}$ , $\mathcal{U}$ is stable if and only if $\mathcal{U}^{\prime}$ is stable.

Fixing $r$ and $c_{1}$ , the Bogomolov inequality implies:

\mathcal{M}^{ss}_{H}(r,c_{1},N)=\emptyset\quad\text{for}\quad N\ll 0.

Thus, there exists a smallest integer $N_{0}$ such that:

\mathcal{M}^{ss}_{H}(r,c_{1},N_{0})\neq\emptyset.

By Lemma 3.12, we have:

Lemma 3.13.

For every $n\geq N_{0}$ ,

\mathcal{M}^{ss}_{H}(r,c_{1},n)\neq\emptyset.

Let $\theta_{0}:=const+2rN_{0}$ . Yoshioka has shown the following results:

Theorem 3.14 (Theorem 0.1 of [33]; Theorem 0.1 of [34]).

When $S$ is a K3 surface and $H$ is a generic ample divisor, $\theta_{0}=0$ . When $S$ is an abelian surface, $\theta_{0}=2$ .

Remark 3.15.

When $S$ is $\mathbb{P}^{2}$ or $\mathbb{P}^{1}\times\mathbb{P}^{1}$ , the constant $\theta_{0}$ was computed by Drezet-Le Potier in [9].

3.4. Perverse coherent sheaves on blow-ups

Now recall the moduli space of perverse coherent sheaves of Nakajima-Yoshioka [25, 24, 26]. Let $o\in S$ be a closed point and $\hat{S}:=Bl_{o}S$ . We denote $p:\hat{S}\to S$ as the projection map and $C:=p^{-1}(o)\cong\mathbb{P}^{1}$ as the exceptional divisor. We denote

\mathcal{O}_{C}(m):=\mathcal{O}(-mC)|_{C}

which is the unique degree $m$ line bundle on the rational curve $C$ for any $m\in\mathbb{Z}$ .

Definition 3.16.

A stable perverse coherent sheaf $E$ on $\hat{S}$ with respect to $H$ is a coherent sheaf such that

Hom(E,\mathcal{O}_{C}(-1))=0,\quad Hom(\mathcal{O}_{C},E)=0

and $p_{*}E$ is Gieseker stable with respect to $H$ .

Given the decomposition of cohomology groups:

H^{*}(\hat{S},\mathbb{Q})=p^{*}(H^{*}(\hat{S},\mathbb{Q}))\oplus\mathbb{Q}[C]

where $[C]\in H^{1,1}(\hat{S})$ and $[C]^{2}=-1$ , we define a morphism

p_{!}:H^{*}(\hat{S},\mathbb{Q})\to H^{*}(S,\mathbb{Q}),\quad p_{!}(c):=p_{*}(c\ td(\hat{S}))td(S)^{-1}.

The morphism $p_{!}$ is compatible with the algebraic $K$ -theory, i.e. we have the commutative diagram

where $ch$ is the Chern character morphism. Moreover, we will have

p_{!}([\hat{S}])=[S],\quad p_{!}([C])=\frac{1}{2}[pt],\quad p_{!}[pt]=pt.

and

p_{!}p^{*}=id:H^{*}(S,\mathbb{Q})\to H^{*}(S,\mathbb{Q}).

Fixing a total Chern class $c=(r,c_{1},n)$ we denote

ch(c):=r+c_{1}+\frac{1}{2}c_{1}^{2}-n

which is the Chern chracter of $c$ .

Definition 3.17.

For any total Chern class $c$ on $S$ and $l\in\mathbb{Z}$ , we define

M^{0}(c,l)

as the moduli space of perverse coherent sheaves with the total Chern chracter

v_{d}:=p^{*}ch(c)-d(ch(\mathcal{O}_{C}(-1)))=p^{*}ch(c)-d([C]-\frac{1}{2}[pts]).

Proposition 3.18 (Lemma 3.22 of [24] and Lemma 3.3 of [19]).

Assuming 3.8 and 3.8, $M^{0}(c,l)$ is either empty or a smooth variety of expected dimension

const+2rn-r(l+r).

Proposition 3.19 (Proposition 3.1 of [20] and Lemma 3.4 of [24]).

For a stable coherent sheaf $E$ on $\hat{S}$ , we have

R^{i}p_{*}E(aC)=0,\quad a=0,1,\quad i\geq 1,

and moreover,

Rp_{*}E(aC)=p_{*}E(aC),\quad a=0,1,

are all stable.

By Proposition 3.19, there exist canonical morphisms

	$\displaystyle\zeta:M^{0}(c,l)\to M^{s}_{H}(c),$	$\displaystyle E\to p_{*}E,$
	$\displaystyle\eta:M^{0}(c,l)\to M^{s}_{H}(c-l[\mathrm{pts}]),$	$\displaystyle E\to p_{*}E(C).$

We denote

\mathcal{U}_{o}\in\mathrm{Coh}(\mathcal{M}^{s}_{H}(c)):=\mathcal{U}|_{\mathcal{M}^{s}_{H}(c)\times o}.

Theorem 3.20 (Theorem 4.1 of [24], also see Theorem 4.1 of [19]).

We have

M^{0}(c,l)\cong Gr_{\mathcal{M}^{s}_{H}(c)}(\mathcal{U}^{\vee}_{o}[1],l),\quad M^{0}(c,l)\cong Gr_{\mathcal{M}^{s}_{H}(c-l[pts])}(\mathcal{U}_{o},l)

where the projection morphisms are just $\zeta$ and $\eta$ respectively.

By Proposition 3.10, Lemma 3.13 and Theorem 3.20, for any $o\in S$ , the family $X_{m}:=\mathcal{M}_{H}^{s}(r,c_{1},m)$ and the coherent sheaf $\mathcal{U}_{o}$ satisfy the conditions in Proposition 3.1 and Corollary 3.3, with the given $const$ and $\theta_{0}$ . Hence by Proposition 3.1 and Corollary 3.3 we have

Corollary 3.21 (Theorem 1.1 and Theorem 1.2).

Let $\theta:=\theta_{0}+2$ , and define

vdim_{n,l}:=\dim(\mathcal{M}_{H}^{s}(r,c_{1},n))+2+\max\{0,l(r-l)\},

for any $n\in\mathbb{Z}$ and $l\geq 0$ . Then, the degeneracy locus

\bar{D}_{\mathcal{U}_{n},l}

is either an empty set or an irreducible Cohen-Macaulay variety of dimension $vdim_{n,l}$ . Moreover, $\bar{D}_{\mathcal{U}_{n},l}\neq\emptyset$ if and only if $\mathcal{M}_{H}^{s}(r,c_{1},n)\neq\emptyset$ and

vdim_{n,l}\geq\theta+t(r-t),

where $t$ is the remainder of $l$ divided by $r$ .

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The Degeneracy Loci for Smooth Moduli of Sheaves

Abstract.

1. Introduction

1.1. The Degeneracy Loci for Smooth Moduli of Sheaves

Theorem 1.1.

Theorem 1.2.

Remark 1.3.

Remark 1.4.

Remark 1.5.

1.2. The Degeneracy Locus and the Grassmannian

Proposition 1.6.

Remark 1.7.

Remark 1.8.

Definition 1.9.

1.3. Incidence Varieties, Birational Geometry, and Derived Algebraic Geometry

Theorem 1.10.

Proposition 1.11 (Proposition 2.17).

1.4. The Virtual Fundamental Class of Incidence Varieties

Proposition 1.12 (Proposition 4.6 of [37]).

1.5. Perverse Coherent Sheaves on Blow-ups and Derived Grassmannians

1.6. Some Notations

1.7. Acknowledgement

2. Degeneracy Theory for Tor-Amplitude [0,1][0,1]-Complexes

2.1. Degeneracy Loci of a Tor-Amplitude [0,1][0,1]-Complex

Definition 2.1 (Section 14.4 of [10]).

Example 2.2.

Proposition 2.3 (Theorem 14.4 (b)(c) of [10]).

Proof.

Definition 2.4.

Lemma 2.5.

Lemma 2.6.

Lemma 2.7.

2.2. Derived Grassmannians of a Tor-Amplitude [0,1][0,1]-Complex

Definition 2.8 (Definition 4.3 of [14]).

Remark 2.9.

Definition 2.10.

Theorem 2.11 ([14]).

Definition 2.12 (Incidence Correspondence, Definition 2.7 of [16]).

2.3. Degeneracy Loci and the Classicality of Incidence Varieties

Theorem 2.13.

Proposition 2.14.

Proposition 2.15.

Lemma 2.16.

Proof.

Proof of Proposition 2.14.

Proof of Proposition 2.15.

2.4. Birational Geometry of Incidence Varieties for Perfect Degeneracy Loci

Proposition 2.17 (Proposition 1.11).

Proof.

3. Degeneracy Loci on the Smooth Moduli Surface of Coherent Sheaves over a Surface

3.1. A General Proposition

Proposition 3.1.

Proof.

Proposition 3.2.

Proof.

Corollary 3.3.

3.2. Moduli Space of Framed Coherent Sheaves on the Plane and ADHM Construction

Theorem 3.4 (Chapter 2 of [23]).

Theorem 3.5 (Nakajima-Yoshioka [25]).

Theorem 3.6 ([25]).

Corollary 3.7.

3.3. Moduli Space of Gieseker Stable Sheaves

Assumption 3.8.

Remark 3.9.

Proposition 3.10 ([13]; see also Proposition 2.10 of [28] and Lemma 3.3 of [19]).

Proposition 3.11 (Proposition 2.14 of [28]).

Lemma 3.12 (Proposition 5.5 of [27]).

Lemma 3.13.

Theorem 3.14 (Theorem 0.1 of [33]; Theorem 0.1 of [34]).

Remark 3.15.

3.4. Perverse coherent sheaves on blow-ups

Definition 3.16.

Definition 3.17.

Proposition 3.18 (Lemma 3.22 of [24] and Lemma 3.3 of [19]).

Proposition 3.19 (Proposition 3.1 of [20] and Lemma 3.4 of [24]).

Theorem 3.20 (Theorem 4.1 of [24], also see Theorem 4.1 of [19]).

Corollary 3.21 (Theorem 1.1 and Theorem 1.2).

References

2. Degeneracy Theory for Tor-Amplitude $[0,1]$ -Complexes

2.1. Degeneracy Loci of a Tor-Amplitude $[0,1]$ -Complex

2.2. Derived Grassmannians of a Tor-Amplitude $[0,1]$ -Complex