Emma Brakkee
Motivated by the relation between (twisted) K3 surfaces and special cubic fourfolds, we construct moduli spaces of polarized twisted K3 surfaces of any fixed degree and order. We do this by mimicking the construction of the moduli space of untwisted polarized K3 surfaces as a quotient of a bounded symmetric domain.
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The object of this note is the moduli spaces of cubic fourfolds (resp., Gushel-Mukai fourfolds) which contain some special rational surfaces. Under some hypotheses on the families of such surfaces, we develop a general method to show the unirationality of the moduli spaces of the $n$-pointed such fourfolds. We apply this to some codimension 1 loci of cubic fourfolds (resp., Gushel-Mukai fourfolds) appeared in the literature recently.
We give an interpretation of the Fano variety of lines on a cubic fourfold and of the hyperkahler eightfold, constructed by Lehn, Lehn, Sorger and van Straten from twisted cubic curves in a cubic fourfold non containing a plane, as moduli spaces of Bridgeland stable objects in the Kuznetsov component. As a consequence, we reprove the categorical version of Torelli Theorem for cubic fourfolds, we obtain the identification of the period point of LLSvS eightfold with that of the Fano variety, and we discuss derived Torelli Theorem for cubic fourfolds.
We describe the GIT compactification of the moduli space of cubic fourfolds, with a special emphasis on the role played by singularities. Our main result is that a cubic fourfold with only isolated simple (A-D-E) singularities is GIT stable. Conversely, with some minor exceptions, the stability for cubic fourfolds is characterized by this condition.
In this paper we compute the cohomology of moduli space of cubic fourfolds with ADE type singularities relying on Kirwan's blowup and Laza's GIT construction. More precisely, we obtain the Betti numbers of Kirwan's resolution of the moduli space. Furthermore, by applying decomposition theorem we obtain the Betti numbers of the intersection cohomology of Baily-Borel compactification of the moduli space.
We characterize the image of the period map for cubic fourfolds with at worst simple singularities as the complement of an arrangement of hyperplanes in the period space. It follows then that the GIT compactification of the moduli space of cubic fourfolds is isomorphic to the Looijenga's compactification associated to this arrangement. This work builds on and is a natural continuation of our previous paper on the GIT compactification of the moduli space of cubic fourfolds.
We study lines and twisted cubics on cubic fourfolds with simple isolated singularities. We show that the Hilbert scheme compactification of the total space of Starr's fibration on the space of twisted cubics on a cubic fourfold with simple isolated singularities and not containing a plane admits a contraction to a singular projective symplectic variety of dimension eight. The latter admits a crepant resolution which is a deformation of the symplectic eightfold constructed from the space of twisted cubics on a smooth cubic fourfold. Thereby we give another proof of the fact that the latter is …
The starting point of this note is our recent paper with Laza and Saccà on the construction of deformations of O'Grady's $10$-dimensional manifolds as compactifications of intermediate Jacobian fibrations associated to cubic fourfolds. The note provides a complement to that paper consisting in the analogous construction in the twisted case, leading to isogenous but presumably not isomorphic or birational hyper-Kähler manifolds.
In this paper we realize the moduli spaces of cubic fourfolds with specified automorphism groups as arithmetic quotients of complex hyperbolic balls or type IV symmetric domains, and study their compactifications. Our results mainly depend on the well-known works about moduli space of cubic fourfolds, including the global Torelli theorem proved by Voisin ([Voi86]) and the characterization of the image of the period map, which is given by Laza ([Laz09, Laz10]) and Looijenga ([Loo09]) independently. The key input for our study of compactifications is the functoriality of Looijenga compactificati…
We construct a new twenty-dimensional family of projective eight-dimensional irreducible holomorphic symplectic manifolds: the compactified moduli space M_3(Y) of twisted cubics on a smooth cubic fourfold Y that does not contain a plane is shown to be smooth and to admit a contraction M_3(Y) -> Z(Y) to a projective eight-dimensional symplectic manifold Z(Y). The construction is based on results on linear determinantal representations of singular cubic surfaces.
We show that a cubic fourfold F that is apolar to a Veronese surface has the property that its variety of power sums VSP(F,10) is singular along a K3 surface of genus 20. We prove that these cubics form a divisor in the moduli space of cubic fourfolds and that this divisor is not a Noether-Lefschetz divisor. We use this result to prove that there is no nontrivial Hodge correspondence between a very general cubic and its VSP.
We study the geometry of some moduli spaces of twisted sheaves on K3 surfaces. In particular we introduce induced automorphisms from a K3 surface on moduli spaces of twisted sheaves on this K3 surface. As an application we prove the unirationality of moduli spaces of irreducible holomorphic symplectic manifolds of $K3^{[2]}$-type admitting non symplectic involutions with invariant lattices $U(2)\oplus D_4(-1)$ or $U(2)\oplus E_8(-2)$. This complements the results obtained in [Mongardi and Wandel 2015], [Bossiere et al 2016], and the results from [arXiv:1603.00403] about the geometry of IHS fou…
We apply the methods of C{ă}ld{ă}raru to construct a twisted Fourier-Mukai transform between a pair of holomorphic symplectic four-folds. More precisely, we obtain an equivalence between the derived category of coherent sheaves on a certain four-fold and the derived category of twisted sheaves on its `mirror' partner. As corollaries, we show that the two spaces are connected by a one-parameter family of deformations through Lagrangian fibrations, and we extend the original Fourier-Mukai transform to degenerations of abelian surfaces.
Finite generation of Noether-Lefschetz divisors and the slope of the moduli space of cubic fourfolds
本文研究了立方四重面模空间和K3表面的除子结构，通过引入斜率量和显式界限，揭示了有效锥的性质，并给出了K3模空间有理Picard群的生成方式。
We find explicit projective models of a compact Shimura curve and of a (non-compact) surface which are the moduli spaces of principally polarised abelian fourfolds with an automorphism of order five. The surface has a 24-nodal canonical model in IP^4 which is the complete intersection of two S_5-invariant cubics. It is dominated by a Hilbert modular surface and we give a modular interpretation for this. We also determine the L-series of these varieties as well as those of several modular covers of the Shimura curve.
We revisit the work of Lehn-Lehn-Sorger-van Straten on twisted cubic curves in a cubic fourfold not containing a plane in terms of moduli spaces. We show that the blow-up $Z'$ along the cubic of the irreducible holomorphic symplectic eightfold $Z$, described by the four authors, is isomorphic to an irreducible component of a moduli space of Gieseker stable torsion sheaves or rank three torsion free sheaves. For a very general such cubic fourfold, we show that $Z$ is isomorphic to a connected component of a moduli space of tilt-stable objects in the derived category and to a moduli space of Bri…
A 10-dimensional symplectic moduli space of torsion sheaves on the cubic 4-fold is constructed. It parametrizes the stable rank 2 vector bundles on the hypeplane sections of the cubic 4-fold which are obtained by Serre's construction from normal elliptic quintics. The natural projection to the dual projective 5-space parametrizing the hyperplane sections is a Lagrangian fibration. The symplectic structure is closely related (and conjecturally, is equal) to the quasi-symplectic one, induced by the Yoneda pairing on the moduli space.
Double EPW cubes from twisted cubics on Gushel-Mukai fourfolds
本文首次系统研究 GM 四重簇上的扭曲三次曲线，证明双 EPW 立方体是其 Hilbert 概形的 MRC 商，并给出 O'Grady 猜想的新例子。
A nonspecial divisor in the moduli space of cubic fourfolds via 10-nodal plane sextics
本文通过研究10节点平面六次曲线，发现立方四维流形模空间中存在一个非诺特-勒夫施茨类型的不可约除子。
We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a K3 surface. On the other hand, we show that the Fano varieties are always birational to moduli spaces of twisted stable coherent sheaves on a K3 surface. The moduli spaces of complexes and of sheaves are related by wall-crossing in the derived category of twisted sheaves on the corresponding K3 surface.
Constructions of derived equivalent hyper-Kähler fourfolds
本文研究了超凯勒四维簇的扭曲导出等价，描述了特定$K3^{[2]}$型四维簇导出等价的条件，并给出了Picard秩$\geq 2$的超凯勒流形对的新构造。
Double planes branched in 6 lines give a famous example of K3 surfaces. Their moduli are well understood and related to abelian fourfolds of Weil type. We compare these two moduli interpretations and in particular divisors on the moduli spaces. On the K3 side, this is achieved with the help of elliptic fibrations. We also study the Kuga-Satake correspondence on these special divisors.
Conics on Gushel-Mukai fourfolds, EPW sextics and Bridgeland moduli spaces
本研究将与极一般Gushel-Mukai四簇相关的两类EPW六次曲面等同于其Kuznetsov分支中稳定对象的Bridgeland模空间，肯定回答了Perry-Pertusi-Zhao的问题，并证明了Kuznetsov-Perry的相关猜想。
We provide explicit descriptions of the generic members of Hassett's divisors $\mathcal C_d$ for relevant $18\leq d\leq 38$ and for $d=44$. In doing so, we prove that $\mathcal C_d$ is unirational for $18\leq d\leq 38,d=44$. As a corollary, we prove that the moduli space $\mathcal N_{d}$ of polarized K3 surfaces of degree $d$ is unirational for $d=14,26,38$. The case $d=26$ is entirely new, while the other two cases have been previously proven by Mukai.
Lines, Twisted Cubics on Cubic Fourfolds, and the Monodromy of the Voisin Map
本文研究一般三次四重形Y的Fano直线簇F上的Voisin自映射ψ，证明其单值群为极大，通过探究ψ与LLSvS簇Z上反辛对合不动轨迹的相互作用完成该结论。
We prove that the moduli stacks of marked and labelled Hodge-special Gushel-Mukai fourfolds are isomorphic. As an application, we construct rational maps from the stack of Hodge-special Gushel-Mukai fourfolds of discriminant $d$ to the moduli space of (twisted) degree-$d$ polarized K3 surfaces. We use these results to prove a counting formula for the number of 4-dimensional fibers of Fourier-Mukai partners of very general Hodge-special Gushel-Mukai fourfolds with associated K3 surface, and a lower bound for this number in the case of a twisted associated K3 surface.
For any smooth projective moduli space $M$ of Gieseker stable sheaves on a complex projective K3 surface (or an abelian surface) S, we prove that the Chow motive $\mathfrak{h}(M)$ becomes a direct summand of a motive $\bigoplus \mathfrak{h}(S^{k_i})(n_i)$ with $k_i\leq \dim(M)$. The result implies that finite dimensionality of $\mathfrak{h}(M)$ follows from finite dimensionality of $\mathfrak{h}(S)$. The technique also applies to moduli spaces of twisted sheaves and to moduli spaces of stable objects in ${\rm D}^{\rm b}(S,α)$ for a Brauer class $α\in{\rm Br}(S)$. In a similar vein, we investig…
Using the isomorphism between the moduli space of polarized K3 surfaces of genus 14 and the moduli space of special cubic fourfolds of discriminant 26, we establish the rationality of the universal K3 surface of genus 14. Precisely, we show that the universal K3 surface of genus 14 is a projective bundle over a certain moduli space of nodal scrolls in P^5, whose rationality we prove using a degenerate version of Mukai's structure theorem for curves of genus 8.
We prove that every Hassett's Noether-Lefschetz divisor of special cubic fourfolds contains a union of three codimension-two subvarieties, parametrizing rational cubic fourfolds, in the moduli space of smooth cubic fourfolds.
Motivated by the question of rationality of cubic fourfolds, we show that a cubic X has an associated K3 surface in the sense of Hassett if and only if the variety F of lines on X is birational to a moduli space of sheaves on a K3 surface, but that having F birational to Hilb^2(K3) is more restrictive. We compare the loci in the moduli space of cubics where each condition is satisfied.
We prove a reconstruction theorem à la Calabrese-Groechenig for the moduli space parametrizing skyscraper sheaves on a smooth projective variety when these are considered as a system of points in the dg category of perfect complexes on the variety, as axiomatized by Toën and Vaquié. This result is then used to show that, for a cubic fourfold $Y\subset \mathbb{P}^5_\mathbb{C}$, the Kuznetsov category $\mathcal{A}_Y$ is geometric (possibly twisted) if and only if a dg enhancement $T_Y$ of $\mathcal{A}_Y$ admits a system of points whose associated moduli space is a (possibly twisted) K3 surface.
Lagrangian families of Bridgeland moduli spaces from Gushel-Mukai fourfolds
本文证明Gushel-Mukai簇与其超平面截面的Kuznetsov分量间拉回与推出函子保持Bridgeland稳定性，并由此构造GM四维簇模空间的8维Lagrangian族及证明GM三维簇模空间的射影性。
We prove that a very general cubic fourfold containing a plane can be embedded into a holomorphic symplectic eightfold as a Lagrangian submanifold. We construct the desired holomorphic symplectic eightfold as a moduli space of Bridgeland stable objects in the derived category of the twisted K3 surface corresponding to the cubic fourfold containing a plane.
We present some applications of the Macaulay2 software package SpecialFanoFourfolds, a package for working with Hodge-special cubic fourfolds and Hodge-special Gushel--Mukai fourfolds. In particular, we show how to construct new examples of such fourfolds, some of which turn out to be rational. We also describe how to calculate K3 surfaces associated with cubic or Gushel-Mukai fourfolds, which relies on an explicit unirationality of some moduli spaces of K3 surfaces.
We propose two conjectures on a moduli theoretic approach to constructing Lagrangian subvarieties of hyperkähler varieties arising from the Kuznetsov components of cubic fourfolds or Gushel--Mukai fourfolds. Then we verify the conjectures in several cases, recovering classical examples. As a corollary, we confirm a conjecture of O'Grady in several instances on the existence of Lagrangian covering families for hyperkähler varieties.
On the irrationality of cubic fourfolds
该研究证明了立方四重曲面的原始上同调群与K3曲面的中间上同调群同构，揭示了其无理性特征。
We study how the degrees of irrationality of moduli spaces of polarized K3 surfaces grow with respect to the genus $g$. We prove that the growth is bounded by a polynomial function of degree $14+\varepsilon$ for any $\varepsilon>0$ and, for three sets of infinitely many genera, the bounds can be refined to polynomials of degree $10$. The main ingredients in our proof are the modularity of the generating series of Heegner divisors due to Borcherds and its generalization to higher codimensions due to Kudla, Millson, Zhang, Bruinier, and Westerholt-Raum. For special genera, the proof is also buil…
A note on cubic fourfolds containing several planes
研究包含多个平面（特别是沿直线相交的两个平面）的四次三维簇的几何、Hodge理论和导出范畴，并关联到扭曲K3曲面。
Moduli of sheaves on fourfolds as derived Lagrangian intersections
本文证明 $(-2)$-移位辛导出概形可局部表示为光滑经典概形的 $(-1)$-移位余切栈中两个 Lagrangian 的导出交，并讨论其对 CY 四维流形层模栈的意义。
Cubic fourfolds containing highly singular hyperplane sections
本研究构造了五个不可约除子，用于参数化含高度奇异超平面截面的光滑立方四重面，并证明这些除子不是Noether-Lefschetz除子。
We give explicit descriptions of some Noether-Lefschetz divisors in the moduli space of Gushel-Mukai fourfolds. As a consequence we obtain that their Kodaira dimension is negative.
We describe an Azumaya algebra on the resolution of singularities of the double cover of a plane ramified along a nodal sextic associated to a non generic cubic fourfold containing a plane. We show that the derived category of such a resolution, twisted by the Azumaya algebra, is equivalent to the Kuznetsov component in the semiorthogonal decomposition of the derived category of the cubic fourfold.
Moduli of Cubic fourfolds and reducible OADP surfaces
本文研究Hassett除子$\mathcal C_8$与$\mathcal{C}_{12}$、$\mathcal{C}_{20}$的交集不可约分量，刻画一般元素的几何，并借助二次纤维化有理截面和可约OADP曲面探讨其有理性，最终用Macaulay计算给出显式方程。
We prove that the K-moduli space of cubic fourfolds is identical to their GIT moduli space. More precisely, the K-(semi/poly)stability of cubic fourfolds coincide to the corresponding GIT stabilities, which was studied in detail by Laza. In particular, this implies that all smooth cubic fourfolds admit Kähler-Einstein metrics. Key ingredients are local volume estimates in dimension three due to Liu-Xu, and Ambro-Kawamata's non-vanishing theorem for Fano fourfolds.
Moduli spaces on Kuznetsov components are Irreducible Symplectic Varieties
本文研究立方四胞体Kuznetsov分支中具非本原非OG10型Mukai向量的Bridgeland半稳定对象模空间，证明其为具因子终端奇点的射影不可约辛簇，形变类由m和v₀²唯一确定，推广并拓展了此前相关模空间的研究结果。
Cubic fourfolds of discriminant 24 and rationality
本文研究判别式24的三次四重簇，通过其扭曲曲面中Brauer类消失的条件，证明了有理性质，并构造了可数无穷多个新的有理例子。
We prove that the moduli space of cubic fourfolds $\mathcal{C}$ contains a divisor $\mathcal{C}_{42}$ whose general member has a unirational parametrization of degree 13. This result follows from a thorough study of the Hilbert scheme of rational scrolls and an explicit construction of examples. We also show that $\mathcal{C}_{42}$ is uniruled.
Cubic fourfolds with symplectic automorphisms
本文通过分类特定群的表示，确定具有辛自同构的立方四维流形的射影方程及模空间的不可约分量数量，并探讨了其定义域。
A special cubic fourfold is a smooth hypersurface of degree three and dimension four that contains a surface not homologous to a complete intersection. Special cubic fourfolds give rise to a countable family of Noether-Lefschetz divisors C_d in the moduli space C of smooth cubic fourfolds. These divisors are irreducible 19-dimensional varieties birational to certain orthogonal modular varieties. We use the "low-weight cusp form trick" of Gritsenko, Hulek, and Sankaran to obtain information about the Kodaira dimension of C_d. For example, if d = 6n + 2, then we show that C_d is of general type …
We exhibit new examples of rational cubic fourfolds, parametrized by a countably infinite union of codimension-two subvarieties in the moduli space. Our examples are fibered in sextic del Pezzo surfaces over the projective plane; they are rational whenever the fibration has a rational section.
We exhibit explicit examples of very general special cubic fourfolds with discriminant $d$ admitting an associated (twisted) K3 surface, which have non-isomorphic Fourier-Mukai partners. In particular, in the untwisted setting, we show that the number of Fourier-Mukai partners for a very general special cubic fourfold with discriminant $d$ and having an associated K3 surface, is equal to the number $m$ of Fourier-Mukai partners of its associated K3 surface, if $d \equiv 2 (\text{mod}\,6)$; else, if $d \equiv 0 (\text{mod}\,6)$, the cubic fourfold has $\lceil m/2 \rceil$ Fourier-Mukai partners.
Kodaira dimension of moduli of special $K3^{[2]}$-fourfolds of degree 2
该研究针对次数为2的$K3^{[2]}$-四形模空间的Noether-Lefschetz轨迹，计算了除有限个判别式外对应特殊除子的Kodaira维数，证明多数情形下该空间为一般型代数簇。
In analogy to the case of cubic fourfolds, we discuss the conditions under which the double cover $\tilde{Y}_A$ of the EPW sextic hypersurface associated to a Gushel-Mukai fourfold is birationally equivalent to a moduli space of (twisted) stable sheaves on a K3 surface. In particular, we prove that $\tilde{Y}_A$ is birational to the Hilbert scheme of two points on a K3 surface if and only if the Gushel-Mukai fourfold is Hodge-special with discriminant $d$ such that the negative Pell equation $\mathcal{P}_{d/2}(-1)$ is solvable in $\mathbb{Z}$.
Lagrangian tens of planes, Enriques surfaces and holomorphic symplectic fourfolds
本文研究Enriques曲面Fano模型产生的P^5中十个相互相交平面的模空间，建立其与含该十元组的三次四维流形、Beauville–Donagi全纯辛四维流形及双重EPW六次超曲面等十维模空间的联系，并由此证明带超标记Enriques曲面模空间的不可约性。
GIT stable cubic threefolds and certain fourfolds of $K3^{[2]}$-type
本文研究光滑与奇异节点三次三维簇与 $K3^{[2]}$ 型超凯勒四维簇模空间之间的同构在节点超平面上的退化行为，推广至 $A_i$ 型奇点情形，并引入 $K$ 型凯勒锥截面概念以处理 $A_2$ 情形。
Generalizing a recent construction of Yang and Yu, we study to what extent one can intersect Hassett's Noether-Lefschetz divisors $\mathcal{C}_d$ in the moduli space of cubic fourfolds $\mathcal{C}$. In particular, we exhibit arithmetic conditions on 20 indexes $d_1,\dots, d_{20}$ that assure that the divisors $\mathcal{C}_{d_1},\dots,\mathcal{C}_{d_{20}}$ all intersect one another. This allows us to produce examples of rational cubic fourfolds with an associated K3 surface with rank 20 Néron-Severi group, i.e. a singular K3 surface.
We classify the symplectic automorphism groups for cubic fourfolds. The main inputs are the global Torelli theorem for cubic fourfolds and the classification of the fixed-point sublattices of the Leech lattice. Among the highlights of our results, we note that there are 34 possible groups of symplectic automorphisms, with 6 maximal cases. The six maximal cases correspond to 8 non-isomorphic, and isolated in moduli, cubic fourfolds; six of them previously identified by other authors. Finally, the Fermat cubic fourfold has the largest possible order (174,960) for the automorphism group (non-nece…
In this note we introduce the transcendental part $t(X)$ of the motive of a cubic fourfold $X$ and prove that it is isomorphic to the (twisted) transcendental part $h_2^{tr}(F(X))$ in a suitable Chow-Künneth decomposition for the motive of the Fano variety of lines $F(X)$. Then we prove that $t(X)$ is isomorphic to the Prym motive associated to the surface $S_l \subset F(X)$ of lines meeting a general line $l$. If $X$ is a special cubic fourfold in the sense of Hodge theory, and $F(X)\cong S^{[2]}$, with $S$ a $K3$, then we show that $t(X) \cong t_2(S)(1)$, where $t_2(S)$ is the transcendental…
Let $Y$ be the del Pezzo $4$-fold defined by the linear section $\textrm{Gr}(2,5)$ by $\mathbb{P}^7$. In this paper, we classify the type of normal bundles of lines in $Y$ and describe its parameter space. As a corollary, we obtain the desigularized model of the moduli space of stable maps in $Y$. Also we compute the intersection Poincaré polynomial of the stable maps space.
We study ACM bundles on cubic fourfolds containing a plane exploiting the geometry of the associated quadric fibration and Kuznetsov's treatment of their bounded derived categories of coherent sheaves. More precisely, we recover the K3 surface naturally associated to the fourfold as a moduli space of Gieseker stable ACM bundles of rank four.
In [1309.1899], Ranestad and Voisin showed, quite surprisingly, that the divisor in the moduli space of cubic fourfolds consisting of cubics "apolar to a Veronese surface" is not a Noether-Lefschetz divisor. We give an independent proof of this by exhibiting an explicit cubic fourfold X in the divisor and using point counting methods over finite fields to show X is Noether-Lefschetz general. We also show that two other divisors considered in [ibid.] are not Noether-Lefschetz divisors.
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