We find generators of the group of birational automorphisms of the
Hessian surface of a general cubic surface. Its nonsingular minimal model is a K3 surface with the Picard lattice of rank 16. The latter embeds naturally in the even unimodular lattice $II^{1,25}$ of rank 26 and signature $(1,25)$ as the orthogonal complement of a root sublattice of rank 10. Our generators are related to reflections with respect to some Leech roots. A similar observation was made first in the case of quartic Kummer surfaces in the work of S. Kond$\bar {\roman o}$. We shall explain how our generators are related to the generators of the group of birational automorphisms of a general quartic Kummer surface which is birationally isomorphic to a special Hessian surface.
In this article, we study birational transformations belonging to Galois points for certain plane quartic curve. In fact, we see that they can be extended to Cremona transformations. In particular, we determine the conjugacy class of them. We have that they are all conjugate to linear transformations.
The paper explores the birational geometry of terminal quartic 3-folds. In doing this I develop a new approach to study maximal singularities with positive dimensional centers. This allows to determine the pliability of a Q-factorial quartic with ordinary double points, and it shows the importance of Q-factoriality in the context of birational geometry of uniruled 3-folds
It is well-known that a nonsingular minimal cubic surface is birationally rigid; the group of its birational selfmaps is generated by biregular selfmaps and birational involutions such that all relations between the latter are implied by standard relations between reflections on an elliptic curve. It is also known that a factorial nodal quartic threefold is birationally rigid and its group of birational selfmaps is generated by biregular ones and certain birational involutions. We prove that all relations between these involutions are implied by standard relations on elliptic curves, complete …
This is the unabridged web version of the paper that will be published on the American Journal of Mathematics. In this paper, we study the birational geometry of certain examples of mildly singular quartic 3-folds. A quartic 3-fold is an example of a Fano variety, that is, a variety $X$ with ample anticanonical sheaf $Ø_X(-K_X)$. From the point of view of birational geometry they basically fall within two classes: either $X$ is ``close to being rational'', and then it has very many biregularly distinct birational models as a Fano 3-fold, or, at the other extreme, $X$ has a unique model and it …
We construct a determinantal family of quarto-quartic transformations of a complex projective space of dimension $3$ from trigonal curves of degree $8$ and genus $5$. Moreover we show that the variety of $(4,4)$-birational maps of $\mathbb{P}_3(\mathbb{C})$ has at least four irreducible components and describe three of them.
Two birational subvarieties of P^n are called Cremona equivalent if there is a Cremona modification of P^n mapping one to the other. If the codimension of the varieties is at least 2 then they are always Cremona Equivalent. For divisors the question is much more subtle and a general answer is unknown. In this paper I study the case of rational quartic surfaces and prove that they are all Cremona equivalent to a plane.
Three-dimensional del Pezzo varieties of degree 2 are double covers of projective space $\mathbb{P}^{3}$ branced in a quadric. In this paper we prove that if a del Pezzo variety of degree 2 has exactly 15 nodes then the corresponding quadric is a hyperplane section of the Igusa quartic or, equivalently, all such del Pezzo varieties are members of one particular linear system on the Coble fourfold. Their automorphism groups are induced from the automorphism group of Coble fourfold. Also we classify all $G$-birationally rigid varieties of such type.
We shall show the existence of 15 automorphic forms of weight 8 on the moduli space of marked Hessian quartic surfaces of cubic surfaces. These automorphic forms can be interpreted in terms of the coefficients of the Sylvester form of a general cubic surface.
In 1899, Hutchinson presented a way to obtain a three-parameter family of Hessians of cubic surfaces as blowups of Kummer surfaces. We show that this family consists of those Hessians containing an extra class of conic curves. Based on this, we find the invariant of a cubic surface C in pentahedral form that vanishes if its Hessian is in Hutchinson's family, and we give an explicit map between cubic surfaces in pentahedral form and blowups of Kummer surfaces.
We explain a classical construction of a del Pezzo surface of degree d = 4 or 5 as a smooth order two congruence of lines in 3-space whose focal surface is a quartic surface $X_{20-d}$ with 20-d ordinary double points. We also show that $X_{15}$ can be realized as a hyperplane section of the Castelnuovo-Richmond-Igusa quartic hypersurface. This leads to the proof of rationality of the moduli space of 15-nodal quartic surfaces. We discuss some other birational models of $X_{15}$: quartic symmetroids, 5-nodal quartic surfaces, 10-nodal sextic surfaces in $P^4$ and nonsingular surfaces of degree …
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.
This is a continuation of [Og12], concerning automorphisms of smooth quartic K3 surfaces and birational automorphisms of ambient projective three spaces.
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…
For every smooth quartic threefold, we classify all pencils on it whose general element is an irreducible surface birational to a smooth surface of Kodaira dimension zero.
Let Y be a complex Enriques surface whose universal cover X is birational to a general quartic Hessian surface. Using the result on the automorphism group of X due to Dolgachev and Keum, we obtain a finite presentation of the automorphism group of Y. A fundamental domain of the action of the automorphism group on the nef cone of Y is described explicitly. The list of elliptic fibrations on Y and the list of combinations of rational double points that can appear on a surface birational to Y are presented. As an application, a set of generators of the automorphism group of the generic Enriques s…
Motivated by a kind of Penrose correspondence, we investigate the space of hyperplane sections of Segre quartic surfaces which have an ordinary cusp. We show that the space of such hyperplane sections is empty for two kinds of Segre surfaces, and it is a connected surface for all other kinds of Segre surfaces. We also show that when it is non-empty, the closure of the space is either birational to the surface itself or birational to a double covering of the surface, whose branch divisor consists of some specific lines on the surface.
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…
In this paper we study the Hessian map $h_{d,r}$ which associates to any hypersurface of degree $d$ in ${\mathbb P}^r$ its Hessian hypersurface. We study general properties of this map and we prove that: $h_{d,1}$ is birational onto its image if $d\geq 5$; we study in detail the maps $h_{3,1}$, $h_{4,1}$ and $h_{3,2}$; we study the restriction of the Hessian map to the locus of hypersurfaces of degree $d$ with Waring rank $r+2$ in ${\mathbb P}^r$, proving that this restriction is injective as soon as $r\geq 2$ and $d\geq 3$, which implies that $h_{3,3}$ is birational onto its image; we prove t…
Two birational projective varieties in $P^n$ are Cremona Equivalent if there is a birational modification of $P^n$ mapping one onto the other. The minimal Cremona degree of $X\subset P^n$ is the minimal integer among all degrees of varieties that are Cremona Equivalent to $X$. The Cremona Equivalence and the minimal Cremona degree is well understood for subvarieties of codimension at least $2$ while both are in general very subtle questions for divisors. In this note I compute the minimal Cremona degree of quartic surfaces in $P^3$. This allows me to show that any quartic surface of elliptic r…
We prove birational rigidity and calculate the group of birational automorphisms of a nodal Q-factorial double cover $X$ of a smooth three-dimensional quadric branched over a quartic section. We also prove that $X$ is Q-factorial provided that it has at most 11 singularities; moreover, we give an example of a non-Q-factorial variety of this type with 12 simple double singularities.
We classify quartic del Pezzo surface fibrations over the projective line via numerical invariants, giving explicit examples for small values of the invariants. For generic such fibrations, we describe explicitly the geometry of spaces of sections to the fibration, and mappings to the intermediate Jacobian of the total space. We exhibit examples where these are birational, which has applications to arithmetic questions, especially over finite fields.
Mutations of the cluster variables generating the cluster algebra of type $A^{(2)}_2$ reduce to a two-dimensional discrete integrable system given by a quartic birational map. The invariant curve of the map is a singular quartic curve, and its resolution of the singularity induces a discrete integrable system on a conic governed by a cubic birational map conjugate to the cluster mutations of type $A^{(2)}_2$. Moreover, it is shown that the conic is also the invariant curve of the quadratic birational map arising from the cluster mutations of type $A^{(1)}_1$ and the two birational maps on the …
We describe a set of generators and defining relations for the group of birational automorphisms of a general 15-nodal quartic surface in the complex projective 3-dimensional space.
We study two rational Fano threefolds with an action of the icosahedral group $\mathfrak{A}_5$. The first one is the famous Burkhardt quartic threefold, and the second one is the double cover of the projective space branched in the Barth sextic surface. We prove that both of them are $\mathfrak{A}_5$-Fano varieties that are $\mathfrak{A}_5$-birationally superrigid. This gives two new embeddings of the group $\mathfrak{A}_5$ into the space Cremona group.
The Hessian of a general cubic surface is a nodal quartic surface, hence its desingularisation is a K3 surface. We determine the transcendental lattice of the Hessian K3 surface for various cubic surfaces (with nodes and/or Eckardt points for example). Classical invariant theory shows that the moduli space of cubic surfaces is a weighted projective space. We describe the singular locus and some other subvarieties of the moduli space.
Consider a cubic surface satisfying the mild condition that it may be described in Sylvester's pentahedral form. There is a well-known Enriques or Coble surface S with K3 cover birationally isomorphic to the Hessian surface of this cubic surface. We describe the nef cone and the (-2)-curves of S. In the case of pentahedral parameters (1, 1, 1, 1, nonzero t) we compute the automorphism group of S. For t not 1 it is the semidirect product of the free product (Z/2)*(Z/2)*(Z/2)*(Z/2) by the symmetric group S4. In the special case t=1/16 we study the action of Aut(S) on an invariant smooth rational…
Looijenga has introduced new compactifications of locally symmetric varieties that give a complete understanding of the period map from the GIT moduli space of plane sextics to the Baily-Borel compactification of the moduli space polarized K3's of degree 2, and also of the period map of cubic fourfolds. On the other hand, the period map of the GIT moduli space of quartic surfaces is significantly more subtle. In our previous paper, we introduced a Hassett-Keel-Looijenga program for certain locally symmetric varieties of Type IV. As a consequence, we gave a complete conjectural decomposition in…
We prove that the moduli space X(1,7) of (1,7)-polarized abelian surfaces with canonical level structure is birational to the Fano 3-fold V22 of polar hexagons of the Klein quartic X(7). In particular X(1,7) is rational and the birational map to P^3 is defined over Q. As a byproduct we obtain explicitely the equations of the (1,7)-very-ample polarized abelian surfaces in P^6.
In previous work, we have introduced a program aimed at studying the birational geometry of locally symmetric varieties of Type IV associated to moduli of certain projective varieties of K3 type. In particular, a concrete goal of our program is to understand the relationship between GIT and Baily-Borel compactifications for quartic K3 surfaces, K3's which are double covers of a smooth quadric surface, and double EPW sextics. In our first paper (arXiv:1607.01324), based on arithmetic considerations, we have given conjectural decompositions into simple birational transformations of the period ma…
Let $Z$ be the quotient of the Siegel modular threefold $\mathcal{A}^{\rm sa}(2,4,8)$ which has been studied by van Geemen and Nygaard. They gave an implication that some 6-tuple $F_Z$ of theta constants which is in turn known to be a Klingen type Eisenstein series of weight 3 should be related to a holomorphic differential $(2,0)$-form on $Z$. The variety $Z$ is birationally equivalent to the tangent cone of Fermat quartic surface in the title. In this paper we first compute the L-function of two smooth resolutions of $Z$. One of these, denoted by $W$, is a kind of Igusa compactification such…
We introduce one of the most beautiful algebraic varieties known, a quintic hypersurface in projective five-space, which is invariant under the action of the Weyl group of $E_6$. This variety is intricately related with many other moduli problems, some of which are: marked hyperelliptic curves of genus two, Picard curves of genus four with a $\sqrt{-3}$-level structure, six points on the projective line, abelian surfaces with (1,3) polarisations, quartic surfaces invariant under the action of the Heisenberg group in projective three-space, K3-surfaces which are double covers of the projective …
