Special Cubic Fourfolds

By Brendan Hassett. Compositio Mathematica, 2000

286 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/d8cb1b921727362dcdc72363873e5c81ec5b8fcb

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Citations in the corpus

The integral Hodge conjecture for two-dimensional Calabi–Yau categories (2020, background): “(In fact the Hodge diamond of X can be computed explicitly, see [33, 23], but the following argument gives a simpler proof of the Tate type statement.”
Derived Categories of Cubic Fourfolds (2008, background): “(1) Pfaffian cubic fourfolds [Tr1, Tr2]; (2) cubic fourfolds Y containing a plane P ∼= P2 and a 2-dimensional algebraic cycle T such that T � H2 − T � P is odd [ Ha1 , Ha2];”
Homological projective duality for Grassmannians of lines (2006, background): “8 ⊂ (P14)∨ with the Grassmannian Gr(2,6). A relation of cubic 4-folds to K3-surfaces has been known for a long time. For example, they have similar primitive Hodge structures in the middle cohomology [H2]. Also, it is known that the Fano variety of lines on a cubic 4-fold is a deformation of the Hil”
Universal Unramified Cohomology of Cubic Fourfolds Containing a Plane (2013, methodology): “Hassett [36] describes, via lattice theory, Noether–Lefschetz divisors Cd in the moduli space C of cubic fourfolds.”
On the universal $CH_0$ group of cubic hypersurfaces (2014, background): “In dimension 4, we get further precise consequences, for example we prove that a cubic fourfold which is special in the sense of Hassett [15], with discriminant not divisible by 4, has universally trivial CH0 group.”
On two rationality conjectures for cubic fourfolds (2014, background): “Hassett [8, Prop.”
The moduli space of cubic fourfolds (2007, background): “Hassett [14, §4.4] has noticed that the point ω corresponding to the secant variety of the Veronese surface plays a special role in the Hodge theoretical analysis of cubic fourfolds.”
Hodge theory and derived categories of cubic fourfolds (2012, unknown): “m cohomology H(S). This means we can often reduce deformation/obstruction theory questions to Hodge theory. We are guided by the deformation theory 4Note that this is not what Hassett calls Cmar d in [19]; his is a quotient of ours by the group Gbelow. HODGE THEORY AND DERIVED CATEGORIES OF CUBIC FO”
Derived Categories View on Rationality Problems (2015, background): “rpose as the notion of a Griﬃths component, but in a slightly diﬀerent way. Second, historically the ﬁrst conjectural characterization of rationality of cubic fourfolds was suggested by B. Hassett in [Has00] in terms of Hodge structures, see also a discussion in [Add15]. The relation between the Con”
The moduli space of cubic fourfolds via the period map (2007, background): “Hassett [21] has studied the period map of cubic fourfolds in relation to the rationality question.”
The K3 category of a cubic fourfold (2015, methodology): “In [18] Hassett showed that in the moduli space C of smooth cubics, the set of those cubics X for which there exists a primitive positive plane Kd Ă H2,2pX,Zq of discriminant d containing the class c1pOp1qq(2) is an irreducible divisor Cd Ă C.”
Remarks on degenerations of hyper-K\"ahler manifolds (2017, methodology): “In fact the easiest specializations of a cubic fourfold linking cubic fourfolds to K3 surfaces are specializations to nodal cubic fourfolds (the divisor C6 in Hassett’s notation) or degenerations to the cubic secant to the Veronese surface in P5 (which give the divisor C2; see [49]).”
The period map for cubic fourfolds (2007, methodology): “Remark 6.4 Hassett considered in [ 4 ] (especially his Theorem 4.1.1) ‘linear’ one parameter smoothings of this fourfold (i.e., given by an equation of the form F + tG).”
Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics (2005, background): “hat XA ∼= X? More generally we may ask the same question for an (X′,H′) which is a limit of (X,H) as above. A similar question for the Fano variety of lines on a cubic 4-fold is studied by Hassett in [6]. 7.1 Anotherdescription of YA Let V be a 6-dimensional complex vector-space. Let LG(∧3V)† ⊂ LG(∧”
Special prime Fano fourfolds of degree 10 and index 2 (2013, methodology): “d (in the sense of Proposition 6.6) and they are even birationally isomorphic. In §8, we characterize the positive integers d for which there exist (smooth) special fourfolds of discriminant d. As in [H1], our construction relies on the surjectivity of the period map for K3 surfaces. Finally, we sum”
Lectures on K3 Surfaces (2016, background): “The following two examples explain the lattice theory behind a curious relation between rationality of cubic fourfolds Y ⊂ P5 and K3 surfaces as suggested by Hassett in [238].”
A categorical invariant for cubic threefolds (2009, background): “By [10], there exists a codimension one subvariety in the moduli space of cubic fourfolds containing a plane consisting of rational cubic fourfolds.”