The depth of a vector bundle E over the projective plane P^2 is the largest integer h such that [E]/h is in the Grothendieck group of coherent sheaves on P^2 where [E] is the class of E in this Grothendieck group. We show that a moduli space of vector bundles is birational to a suitable number of h by h matrices up to simultaneous conjugacy where h is the depth of the vector bundles classified by the moduli space. In particular, such a moduli space is a rational variety if h <= 4 and is stably rational when h divides 420.
We study birational geometry of the moduli space of stable sheaves on a quadric surface with Hilbert polynomial $5m + 1$ and $c_{1} = (2, 3)$. We describe a birational map between the moduli space and a projective bundle over a Grassmannian as a composition of smooth blow-ups/downs.
We investigate the birational geometry (in the sense of Mori's program) of the moduli space of rank 2 semistable parabolic vector bundles on a rational curve. We compute the effective cone of the moduli space and show that all birational models obtained by Mori's program are also moduli spaces of parabolic vector bundles with certain parabolic weights.
We describe a close relation between wall crossings in the birational geometry of moduli space of Gieseker stable sheaves $M_H(v)$ on $\bb{P}^2$ and mini-wall crossings in the stability manifold $Stab(D^b(\bb{P}^2))$.
Let αbe a Schur root; let h=hcf_v(α(v)) and let p = 1 - < α/h,α/h >. Then a moduli space of representations of dimension vector αis birational to p h by h matrices up to simultaneous conjugacy. Therefore, if h=1,2,3 or 4, then such a moduli space is a rational variety and if h divides 420 it is a stably rational variety.
In this paper, we classify Fano manifolds with elementary contractions of birational type such that the second or third exterior power of tangent bundles are numerically effective.
Using wall-crossing for K3 surfaces, we establish birational equivalence of moduli spaces of stable objects on generic Enriques surfaces for different stability conditions. As an application, we prove in the case of a Mukai vector of odd rank that they are birational to Hilbert schemes. The argument makes use of a new Chow-theoretic result, showing that moduli spaces on an Enriques surface give rise to constant cycle subvarieties of the moduli spaces of the covering K3.
We study birational morphisms between smooth projective surfaces that respect a given Poisson structure, with particular attention to induced birational maps between the (Poisson) moduli spaces of sheaves on those surfaces. In particular, to any birational morphism, we associate a corresponding "minimal lift" operation on sheaves of homological dimension <=1, and study its properties. In particular, we show that minimal lift induces a stratification of the moduli space of simple sheaves on the codomain by open subspaces of the moduli space of simple sheaves on the domain, compatibly with the i…
We study birational projective models of ${\mathcal M}_{2,2}$ obtained from the moduli space of curves with nonspecial divisors. We describe geometrically which singular curves appear in these models and show that one of them is obtained by blowing down the Weierstrass divisor in the moduli stack of ${\mathcal Z}$-stable curves $\overline{\mathcal M}_{2,2}({\mathcal Z})$ defined by Smyth. As a corollary, we prove projectivity of the coarse moduli space $\overline{M}_{2,2}({\mathcal Z})$.
In this paper, we study triples of the form (E, theta, phi) over a compact Riemann Surface, where (E, theta) is a Higgs bundle and phi is a global holomorphic section of the Higgs bundle. Our main result is a description of a birational equivalence which relates geometrically the moduli space of Higgs bundles of rank r and degree d to the moduli space of Higgs bundles of rank (r-1) and degree d.
We study the birational geometry of moduli spaces of semistable sheaves on the projective plane via Bridgeland stability conditions. We show that the entire MMP of their moduli spaces can be run via wall-crossing. Via a description of the walls, we give a numerical description of their movable cones, along with its chamber decomposition corresponding to minimal models. As an application, we show that for primitive vectors, all birational models corresponding to open chambers in the movable cone are smooth and irreducible.
In this paper we consider moduli spaces of coherent systems on an elliptic curve. We compute their Hodge polynomials and determine their birational types in some cases. Moreover we prove that certain moduli spaces of coherent systems are isomorphic. This last result uses the Fourier-Mukai transform of coherent systems introduced by Hernández Ruiperez and Tejero Prieto.
We prove finite generation of the algebra of type A conformal blocks over arbitrary stable curves of any genus. As an application we construct a flat family of irreducible normal projective varieties over the moduli stack of stable pointed curves, whose fiber over a smooth curve is a moduli space of semistable parabolic bundles. This generalizes a construction of a degeneration of the moduli space of vector bundles presented in a recent work of Belkale and Gibney.
This is the authors doctoral thesis written at the Humboldt-University Berlin. It contains material from the three separate papers: "On the Kodaira dimension of the moduli space of nodal curves", "On quotients of $\overline{\mathcal{M}}_{g,n}$ by certain subgroups of $S_n$" and "The moduli space of hyperelliptic curves with marked points".
We consider the space $\mathcal R_{g,S_3}^{S_3}$ of curves with a connected $S_3$-cover, proving that for any odd genus $g\geq 13$ this moduli is of general type. Furthermore we develop a set of tools that are essential in approaching the case of $G$-covers for any finite group $G$.
We prove that the moduli space ${\Cal M}_{g,n}$ of smooth curves of genus $g$ with $n$ marked points is rational for $g=6$ and $1 \le n \le 8$, and it is unirational for $g=8$ and $1 \le n \le 11$, $g=10$ and $1 \le n \le 3$, $g=12$ and $n = 1$.
We give a variant of the Beauville--Narasimhan--Ramanan correspondence for irregular parabolic Higgs bundles with semi-simple irregular part and show that it defines a Poisson isomorphism between certain irregular Dolbeault moduli spaces of a curve and relative Picard bundles of families of ruled surfaces over a curve.
We determine the Kodaira dimension of the moduli space of even spin curves for all genera, with one possible exception: The scheme S_g has negative Kodaira dimension for g<8 and it is of general type for g>8. The Kodaira dimension of S_8 is non-negative.
We study birational geometry of the moduli space of parabolic bundles over a projective line, in the framework of Mori's program. We show that the moduli space is a Mori dream space. As a consequence, we obtain the finite generation of the algebra of type A conformal blocks. Furthermore, we compute the H-representation of the effective cone which was previously obtained by Belkale. For each big divisor, the associated birational model is described in terms of moduli space of parabolic bundles.
Moduli spaces of complete collineations are wonderful compactifications of spaces of linear maps of maximal rank between two fixed vector spaces. We investigate the birational geometry of moduli spaces of complete collineations and quadrics from the point of view of Mori theory. We compute their effective, nef and movable cones, the generators of their Cox rings, and their groups of pseudo-automorphisms. Furthermore, we give a complete description of both the Mori chamber and stable base locus decompositions of the effective cone of the space of complete collineations of the 3-dimensional proj…
In order to study wall crossing formula of Donaldson type invariants on the blown-up plane, Nakajima-Yoshioka constructed a sequence of blow-up/blow-down diagrams connecting the moduli space of torsion free framed sheaves on projective plane, and that on its blow-up. In this paper, we prove that Nakajima-Yoshioka's diagram realizes the minimal model program. Furthermore, we obtain a fully-faithful embedding between the derived categories of these moduli spaces.
A $\mathrm{U}(p,q)$-Higgs bundle on a Riemann surface (twisted by a line bundle) consists of a pair of holomorphic vector bundles, together with a pair of (twisted) maps between them. Their moduli spaces depend on a real parameter $α$. In this paper we study wall crossing for the moduli spaces of $α$-polystable twisted $\mathrm{U}(p,q)$-Higgs bundles. Our main result is that the moduli spaces are birational for a certain range of the parameter and we deduce irreducibility results using known results on Higgs bundles. Quiver bundles and the Hitchin-Kobayashi correspondence play an essential rol…
Let U^{'s}_{L}(n,d) be the moduli space of stable vector bundles of rank $n$ and fixed determinant L of degree d on a nodal curve Y. The moduli space of semistable vector bundles of rank n and degree $d$ will be denoted by U'_Y(n,d). We calculate the Brauer groups of U^{'s}_{L}(n,d)$. We study the question of rationality of $U^{'s}_{L}(n,d)$ and $U'_Y(n,d)$.
In this paper we define a Poisson structure on some moduli spaces related to principal G-bundles on elliptic curves, the simplest example being the moduli space of stable pairs: a vector bundle and its global section. We also study birational morphisms between projective spaces appearing as such moduli spaces.
Let $G$ be a simple simply-connected connected linear algebraic group over $\mathbb{C}$. We proved a $2$-birational Torelli theorem for the moduli space of semistable principal $G$-bundles over a smooth curve of genus $\geq 3$, which says that if two such moduli spaces are $2$-birational then the curves are isomorphic. We also proved a $3$-birational Torelli theorem for the moduli space of stable symplectic parabolic bundles over a smooth curve of genus $\geq 4$.
Moduli spaces of sheaves and Hilbert schemes of points have experienced a recent resurgence in interest in the past several years, due largely to new techniques arising from Bridgeland stability conditions and derived category methods. In particular, classical questions about the birational geometry of these spaces can be answered by using new tools such as the positivity lemma of Bayer and Macrì. In this article we first survey classical results on moduli spaces of sheaves and their birational geometry. We then discuss the relationship between these classical results and the new techniques co…
In this paper we connect degenerations of Fano threefolds by projections. Using Mirror Symmetry we transfer these connections to the side of Landau--Ginzburg models. Based on that we suggest a generalization of Kawamata's categorical approach to birational geometry enhancing it via geometry of moduli spaces of Landau--Ginzburg models. We suggest a conjectural application to Hasset--Kuznetsov--Tschinkel program based on new nonrationality "invariants" we consider --- gaps and phantom categories. We make several conjectures for these invariants in the case of surfaces of general type and quadric…
In this article we consider log canonical pairs which are log-smooth. If the corresponding canonical bundle is pseudo-effective, then we show that any quotient of the orbifold cotangent bundle of the pair has a pseudo-effective determinant. One of the new ingredients in the proof is a generalization of the Bogomolov-McQuillan algebraicity criterion in the context of holomorphic foliations whose minimal slope with respect to a movable class is positive.
We construct a finite dimensional Kaehler manifold with a holomorphic, symplectic circle action whose symplectic reduced spaces may be identified with the tau-vortex moduli spaces (or tau-stable pairs). The Morse theory of the circle action induces natural birational maps between the reduced spaces for different values of tau which in the case of rank two bundles can be canonically resolved in a sequence of blow-ups and blow-downs.
Let $S$ be a birationally ruled surface. We show that the moduli schemes $M_S(r,c_1,c_2)$ of semistable sheaves on $S$ of rank $r$ and Chern classes $c_1$ and $c_2$ are irreducible for all $(r,c_1,c_2)$ provided the polarization of $S$ used satisfies a simple numerical condition. This is accomplished by proving that the stacks of prioritary sheaves on $S$ of fixed rank and Chern classes are smooth and irreducible.
In this paper we study the geometry of GIT configurations of $n$ ordered points on $\mathbb{P}^1$ both from the the birational and the biregular viewpoint. In particular, we prove that any extremal ray of the Mori cone of effective curves of the quotient $(\mathbb{P}^1)^n//PGL(2)$, taken with the symmetric polarization, is generated by a one dimensional boundary stratum of the moduli space. Furthermore, we develop some technical machinery that we use to compute the canonical divisor and the Hilbert polynomial of $(\mathbb{P}^1)^n//PGL(2)$ in its natural embedding, and its group of automorphism…
Following Bayer and Macrì, we study the birational geometry of singular moduli spaces $M$ of sheaves on a K3 surface $X$ which admit symplectic resolutions. More precisely, we use the Bayer-Macrì map from the space of Bridgeland stability conditions $\mathrm{Stab}(X)$ to the cone of movable divisors on $M$ to relate wall-crossing in $\mathrm{Stab}(X)$ to birational transformations of $M$. We give a complete classification of walls in $\mathrm{Stab}(X)$ and show that every birational model of $M$ obtained by performing a finite sequence of flops from $M$ appears as a moduli space of Bridgeland …
We show that for $α\in (2/3, 7/10)$, the log canonical model $\bar M_4(α)$ of the pair $(\bar M_4, αδ)$ is isomorphic to the moduli space $\bar M_4^{hs}$ of h-semistable curves, and that there is a birational morphism $Ξ: \bar M_4^{hs} \to \bar M_4(2/3)$ that contracts the locus of curves $C_1\cup_p C_2$ consisting of genus two curves meeting in a node $p$ such that $p$ is a Weierstrass point of $C_1$ or $C_2$. To obtain this morphism, we construct a compact moduli space $\bar M_{2,1}^{hs}$ of pointed genus two curves that have nodes, ordinary cusps and tacnodes as singularity, and prove that …
Let $C$ be a smooth projective irreducible curve of genus $g$. And let $G_α(n,d,l)$ be the moduli space of $α$ stable pairs of a vector bundle of $\rank n, °d$ and a subspace of $H^0(C,E)$ of $\dim = l $. We find an explicit birational map from $G_α (n, d, n+1)$ to $G_α (1, d, n+1)$ for $C$ general, $1/α\gg 0$ and $g \ge n^2-1$. Because of this and other examples, we conjecture $G_α (a, d, a+z)$ maps birationally to $G_α (z, d, a+z)$ for $1/ α\gg 0$ and $C$ general with $g>2$.
We use categorical method and birational geometry to study moduli spaces of quiver representations. From certain "representable" functor, we construct a birational transformation from the moduli space of representations of one quiver to another new quiver with one vertex less. The dimension vector and the stability for the new moduli are determined functorially. We introduce several relative notions of stability to study such a birational transformation. The essential case is proven to be the usual blow-ups. Moreover, we compare the induced ample divisors for the two moduli. We illustrate this…
We study the birational geometry of the moduli space of complete $n$-quadrics $X$. We exhibit generators for Eff$(X)$ and Nef$(X)$, the cone of effective divisors and the cone of nef divisors, respectively. As a corollary X is a Mori Dream space. Furthermore, we run the MMP on $X$ and find a moduli interpretation for the models induced by the generators of the nef cone. In the case of complete quadric surfaces, we describe all the birational models of $X$ induced by the movable cone and find a moduli interpretation for some of these models.
We find all possible isomorphisms and 3-birational maps (i.e., birational maps which induce an isomorphism between open subsets whose respective complements have codimension at least 3) between moduli spaces of parabolic vector bundles with fixed degree. We prove that every 3-birational map can be described as a composition of tensorization by a fixed line bundle, Hecke transformations, dualization, taking pullback by an isomorphism between the curves and the action of the group of automorphisms of the Jacobian variety of the curve which fix the r-torsion. In particular, we prove a Torelli typ…
This paper studies deformations and birational maps between singular moduli spaces of semistable sheaves with 2-divisible Mukai vectors on K3 surfaces. It is showed that under certain conditions, two such moduli spaces of the same dimension can be connected by deformations and birational maps.
We show that the K-moduli spaces of log Fano pairs $(\mathbb{P}^3, cS)$ where $S$ is a quartic surface interpolate between the GIT moduli space of quartic surfaces and the Baily-Borel compactification of moduli of quartic K3 surfaces as $c$ varies in the interval $(0,1)$. We completely describe the wall crossings of these K-moduli spaces. As the main application, we verify Laza-O'Grady's prediction on the Hassett-Keel-Looijenga program for quartic K3 surfaces. We also obtain the K-moduli compactification of quartic double solids, and classify all Gorenstein canonical Fano degenerations of $\ma…
We give a method to construct stable vector bundles whose rank divides the degree over curves of genus bigger than one. The method complements the one given by Newstead. Finally, we make some systematic remarks and observations in connection with rationality of moduli spaces of stable vector bundles.
We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of Bridgeland. This divisor class varies naturally with the Bridgeland stability condition. For a generic stability condition on a K3 surface, we prove that this class is ample, thereby generalizing a result of Minamide, Yanagida, and Yoshioka. Our result also gives a systematic explanation of the relation between wall-crossing for Bridgeland-stability and the minimal model program for the moduli space. We give three applications of our method for classical moduli spaces of sheaves on a K3 surfa…
Let M_{7,n} be the (coarse) moduli space of smooth curves of genus 7 with n marked points defined over the complex field. We denote by M^1_{7,n;4} the locus of points inside M_{7,n} representing curves carrying a g^1_4. It is classically known that M^1_{7,n;4} is irreducible of dimension 17+n. We prove in this paper that M^1_{7,n;4} is rational for 0<= n <= 11.
We show that the moduli spaces of non-triple cyclic coverings of genus 2 curves and that of even spin curves of genus 2 are birationally isomorphic via the Prym map. We describe the log canonical models of the moduli space of even spin curves of genus 2 and use this to extend the above birational map.
By work of Looijenga and others, one has a good understanding of the relationship between GIT and Baily-Borel compactifications for the moduli spaces of degree 2 K3 surfaces, cubic fourfolds, and a few other related examples. The similar-looking cases of degree 4 K3 surfaces and double EPW sextics turn out to be much more complicated for arithmetic reasons. In this paper, we refine work of Looijenga to allow us to handle these cases. Specifically, in analogy with the so-called Hassett-Keel program for the moduli space of curves, we study the variation of log canonical models for locally symmet…
The known counterexamples to the global Torelli theorem for higher-dimensional hyperkahler manifolds are provided by birational manifolds. We address the question whether two birational hyperkahler manifolds (i.e. irreducible symplectic) manifolds always define non-separated points in the moduli space of marked manifolds. An affirmative answer is given for the cases of Mukai's elementary transformations and birational correspondences which are isomorphic in codimension two. The techniques are applied to show that the moduli spaces of rank two sheaves on a K3 surface are deformation equivalent …
We study certain pencils of del Pezzo surfaces generated by a smooth del Pezzo surface S of degree less or equal to 3 anti-canonically embedded into a weighted projective space P and an appropriate multiple of a hyperplane H. Our main observation is that every minimal model program relative to the morphism lifting such pencil on a suitable resolution of its indeterminacies preserves the open subset P H â A^3. As an application, we obtain projective completions of A^3 into del Pezzo fibrations over P^1 of every degree less or equal to 4. We also obtain completions of A^3 into Mori conic bundles…
The moduli space of cubic threefolds in CP4, with some minor birational modifications, is the Baily-Borel compactification of the quotient of the complex 10-ball by a discrete group. We describe both the birational modifications and the discrete group explicitly.
The notion of d-critical loci was introduced by Joyce in order to give classical shadows of $(-1)$-shifted symplectic derived schemes. In this paper, we discuss birational geometry for d-critical loci, by introducing notions such as `d-critical flips', `d-critical flops', etc. They are not birational maps of the underlying spaces, but rather should be understood as virtual birational maps. We show that several wall-crossing phenomena of moduli spaces of stable objects on Calabi-Yau 3-folds are described in terms of d-critical birational geometry. Among them, we show that wall-crossing diagrams…
It is known that the moduli space of plane quartic curves is birational to an arithmetic quotient of a 6-dimensional complex ball. In this paper, we shall show that there exists a 15-dimensional space of meromorphic automorphic forms on the complex ball which gives a birational embedding of the moduli space of plane quartics with level 2 structure into 14-dimensional projective space. This map coincides with the one given by Coble by using Gopel invariants.
We prove that the forgetful morphism from the moduli space of orthogonal bundles to the moduli space of vector bundles over a smooth curve is an embedding. Our proof relies on an explicit description of a set of generators for the invariants on the representation space of a quiver under the action of a product of classical groups.
We define a categorical birational invariant for minimal geometrically rational surfaces with a conic bundle structure over a perfect field via components of a natural semiorthogonal decomposition. Together with the similar known result on del Pezzo surfaces, this provides a categorical birational invariant for geometrically rational surfaces.
