In this note, which is an addendum to the e-print math.AG/9810121, we prove that the variety VSP(F,10) of presentations of a general cubic form F in 6 variables as a sum of 10 cubes is a smooth symplectic 4-fold, which is deformation equivalent to the Hilbert square of a K3 surface of genus 8 but different from the family of lines on a cubic 4-fold. This provides a new geometric construction of a compact complex symplectic fourfold, different from a Hilbert square of a K3 surface, a generalized Kummer 4-fold, the variety of lines on a cubic 4-fold and the recent examples of O'Grady (see Duke Math. J. 134, no. 1 (2006), 99-137).
We show that the affine cones over a general Fano-Mukai fourfold of genus $g=7$, $8$ and $9$ are flexible. Equivalently, there is an infinitely transitive action of the special automorphism group on such affine cones. In particular, any Mukai fourfold of genus $7,8$ and $9$ is $\Bbb A^2$-cylindrical.
We show that the affine cones over any Fano-Mukai fourfold of genus 10 are flexible; in particular, the automorphism group of such a cone acts highly transitively outside the vertex. Furthermore, any Fano-Mukai fourfold of genus 10, with one exception, admits a covering by open charts isomorphic to the affine four-space.
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.
The automorphism groups of the Fano-Mukai fourfold of genus 10 were studied in our previous paper [arXiv:1706.04926]. In particular, we found in [arXiv:1706.04926] the neutral components of these groups. In the present paper we finish the description of the discrete parts. Up to isomorphism, there are two special Fano--Mukai fourfold of genus 10 with the automorphism groups $GL_2(k)\rtimes\mathbb{Z}/2\mathbb{Z}$ and $(\mathbb{G}_a\times\mathbb{G}m)\rtimes\mathbb{Z}/2\mathbb{Z}$, respectively. For any other Fano-Mukai fourfold $V$ of genus 10 one has $\mathrm{Aut}(V)=\mathbb{G}_m^2\rtimes \math…
The main outcome of this paper is that the variety VSP(F,10) of presentations of a general cubic form F in 6 variables as a sum of 10 cubes is a smooth symplectic 4-fold obtained a deformation of the Hilbert square of a K3 surface of genus 8. After publishing it in Trans. Am. Math. Soc. 353, No.4, 1455-1468 (2001), it was noted to us by Eyal Markman that in Theorem 3.17 we conclude without proof that VSP(F,10) should be the 4-fold of lines on another cubic 4-fold. We correct this in the e-print "Addendum to K3 surfaces of genus 8 and varieties of sums of powers of cubic fourfolds" (math.AG/061…
It is known that the moduli space of smooth Fano-Mukai fourfolds $V_{18}$ of genus $10$ has dimension one. We show that any such fourfold is a completion of $\mathbb{C}^4$ in two different ways. Up to isomorphism, there is a unique fourfold $V_{18}^{\mathrm s}$ acted upon by $\operatorname{SL}_2(\mathbb{C})$. The group $\operatorname{Aut}(V_{18}^{\mathrm s})$ is a semidirect product $\operatorname{GL}_2(\mathbb{C})\rtimes(\mathbb{Z}/2\mathbb{Z})$. Furthermore, $V_{18}^{\mathrm s}$ is a $\operatorname{GL}_2(\mathbb{C})$-equivariant completion of $\mathbb{C}^4$, and as well of $\operatorname{GL}…
We study universal families of stable genus two curves with level structure. Among other things, it is shown that the (1,1) part is spanned by divisor classes, and that there are no cycles of type (2,2) in the third cohomology of the first direct image. Using this, we deduce the Hodge and Tate conjectures hold for these varieties.
Coble type hypersurfaces and hyperkähler fourfolds
Universität des Saarlandes（萨尔兰大学）；Université Paris Cité and Sorbonne Université（巴黎西岱大学和索邦大学）；Universitat de Barcelona（巴塞罗那大学）；Università degli Studi di Milano（米兰大学）
This is a short addendum to a note of Beauville on the subject of the title. We prove an inequality that takes into account the constant part of the Jacobian.
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.
Using the Fredholm setup of [12], we study genus zero (and higher) relative Gromov-Witten invariants with maximum tangency of symplectic log Calabi-Yau fourfolds. In particular, we give a short proof of [23, Conjecture 6.2] that expresses these invariants in terms of certain integral invariants by considering generic almost complex structures to obtain a geometric count. We also revisit the localization calculation of the multiple-cover contributions in [23, Proposition 6.1] and recalculate a few terms differently to provide more details and illustrate the computation of deformation/obstructio…
Using the connection discovered by Hassett between the Noether-Lefschetz moduli space of special cubic fourfolds of discriminant 42 and the moduli space F_{22} of polarized K3 surfaces of genus 22, we show that the universal K3 surface over F_{22} is unirational.
O'Grady conjectured that the Chow group of 0-cycles of the generic fiber of the universal family over the moduli space of polarized K3 surfaces of genus g is cyclic. This so-called generalized Franchetta conjecture has been solved only for low genera where there is a Mukai model (precisely, when g<11 and g=12, 13, 16, 18, 20), by the work of Pavic--Shen--Yin. In this paper, as a non-commutative analog, we study the Franchetta property for families of special cubic fourfolds (in the sense of Hassett) and relate it to O'Grady's conjecture for K3 surfaces. Most notably, by using special cubic fou…
A surgery of a real symplectic manifold $X_{\mathbb R}$ along a real Lagrangian sphere $S$ is a modification of the symplectic and real structure on $X_{\mathbb R}$ in a neigborhood of $S$. Genus 0 Welschinger invariants of two real symplectic $4$-manifolds differing by such a surgery have been related in a previous work in collaboration with N. Puignau. In the present paper, we explore some particular situations where these general formulas greatly simplify. As an application, we complete the computation of genus 0 Welschinger invariants of all del~Pezzo surfaces, and of all $\mathbb R$-minim…
The aim of the present paper is to prove the rationality of the universal family of polarized $ K3 $ surfaces of degree 14. This is achieved by identifying it with the moduli space of cubic fourfolds plus the data of a quartic scroll. The last moduli space is finally proved to be rational since it has a natural structure of $\mathbb P^n$-bundle over a $ k $-stably rational variety with $k \leq n$.
For a general cubic fourfold $X\subset\mathbb{P}^5$ with Fano scheme of lines $F$, we prove a number of properties of the universal family of lines $I\to F$ and various subloci. We first describe the moduli and ramification theory of the genus four fibration $p:I\to X$ and explore its relation to a birational model of $F$ in $I$. The main part of the paper is devoted to describing the locus $V\subset F$ of triple lines, i.e., the fixed locus of the Voisin map $ϕ:F\dashrightarrow F$, in particular proving it is an irreducible projective singular surface of class $21\mathrm{c}_2(\mathcal{U}_F)$ …
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…
We compute characteristic numbers of crepant resolutions of Weierstrass models corresponding to elliptically fibered fourfolds $Y$ dual in F-theory to a gauge theory with gauge group $G$. In contrast to the case of fivefolds, Chern and Pontryagin numbers of fourfolds are invariant under crepant birational maps. It follows that Chern and Pontryagin numbers are independent on a choice of a crepant resolution. We present the results for the Euler characteristic, the holomorphic genera, the Todd-genus, the $L$-genus, the $\hat{A}$-genus, and the curvature invariant $X_8$ that appears in M-theory. …
We study the locus of intermediate Jacobians of cubic threefolds within the moduli space of complex principally polarized abelian fivefolds, and its generalization to arbitrary genus - the locus of abelian varieties with a singular odd two-torsion point on the theta divisor. Assuming that this locus has expected codimension (which we show to be true for genus up to 5), we compute the class of this locus, and of is closure in the perfect cone toroidal compactification, in the Chow, homology, and the tautological ring. We work out the cases of genus up to 5 in detail, obtaining explicit expressi…
We show that the family of 21-dimensional intermediate jacobians of cubic fivefolds containing a given cubic fourfold X is generically an algebraic integrable system. In the proof we apply an integrability criterion, introduced and used by Donagi and Markman to find a similar integrable system over the family of cubic threefolds in X. To enter in the conditions of this criterion, we write down explicitly the known by Beauville and Donagi symplectic structure on the family F(X) of lines on the general cubic fourfold X, and prove that the family of planes on a cubic fivefold containing X is embe…
We study punctual quot-schemes of torsion-free sheaves $E_Y$ on smooth projective curves, surfaces and Calabi--Yau fourfolds via their virtual geometry. Our goal is to give a complete description of the virtual fundamental classes and their tautological integrals. In the fourfold case, we first construct these classes under additional conditions. We use novel methods relying on the wall-crossing of Joyce. Our results include -the dependence of the cobordism classes on the torsion-free sheaf $E_Y$ where $Y$ is a surface, -relations to the previous results in the literature, which addressed the …
The Coble cubics were discovered more than a century ago in connection with genus two Riemann surfaces and theta functions. They have attracted renewed interest ever since. Recently, they were reinterpreted in terms of alternating trivectors in nine variables. Exploring this relation further, we show how the Hilbert scheme of pairs of points on an abelian surface, and also its Kummer fourfold, a very remarkable hyper-Kähler manifold, can very naturally be constructed in this context. Moreover, we explain how this perspective allows us to describe the group law of an abelian surface, in a strik…
