Smooth Quartic Surfaces Research Articles

Invariant smooth quartic surfaces by all finite primitive subgroups of [formula omitted

For each finite primitive subgroup G of PGL4(C), we find all the smooth G-invariant quartic surfaces. We also find all the faithful representations in PGL4(C) of the smooth quartic G-invariant surfaces by the groups: A5,S5,PSL2(F7), A6, Z24⋊Z5 and Z24⋊D10. The primitive representation of these groups is precisely the subgroups of PGL4(C) for which P3 is not G-super rigid. As a byproduct, we show that the smooth quartic surface with the biggest group of projective automorphism is given by {x04+x14+x24+x34+12x0x1x2x3=0} (unique up to projective equivalence).

Quartic surfaces, their bitangents and rational points

Let X be a smooth quartic surface not containing lines, defined over a number field K. We prove that there are only finitely many bitangents to X which are defined over K. This result can be interpreted as saying that a certain surface, having vanishing irregularity, contains only finitely many rational points. In our proof, we use the geometry of lines of the quartic double solid associated to X. In a somewhat opposite direction, we show that on any quartic surface X over a number field K, the set of algebraic points in X(\overeline K) which are quadratic over a suitable finite extension K' of K is Zariski-dense.

800 conics on a smooth quartic surface

We construct an example of a smooth spatial quartic surface that contains 800 irreducible conics.

Deep Learning Gauss\u2013Manin Connections

The Gauss–Manin connection of a family of hypersurfaces governs the change of the period matrix along the family. This connection can be complicated even when the equations defining the family look simple. When this is the case, it is expensive to compute the period matrices of varieties in the family via homotopy continuation. We train neural networks that can quickly and reliably guess the complexity of the Gauss–Manin connection of pencils of hypersurfaces. As an application, we compute the periods of 96% of smooth quartic surfaces in projective 3-space whose defining equation is a sum of five monomials; from the periods of these quartic surfaces, we extract their Picard lattices and the endomorphism fields of their transcendental lattices.

Reconstructing curves from their Hodge classes

Let S be a smooth algebraic surface in {mathbb {P}}^3({mathbb {C}}). Movasati and Sertöz (Rend. Circ. Mat. Palermo 2:1–17, 2020) associate an ideal I_{alpha (C)} to the primitive cohomology class alpha (C) of C in S. We show that the equations of C can be determined by I_{alpha (C)} under numerical conditions. We apply this result to reconstruct rational curves and arithmetically Cohen-Macaulay curves from their cohomology classes. On the other hand, we show that the class alpha (C) of a rational quartic curve C on a smooth quartic surface S is not even perfect, that is, that I_{alpha (C)} is bigger than the sum of the Jacobian ideal of S and of the homogeneous ideals of curves D in S for which I_{alpha (D)}=I_{alpha (C)}.

Lines on K3 quartic surfaces in characteristic 3

We investigate the number of straight lines contained in a K3 quartic surface X defined over an algebraically closed field of characteristic 3. We prove that if X contains 112 lines, then X is projectively equivalent to the Fermat quartic surface; otherwise, X contains at most 67 lines. We improve this bound to 58 if X contains a star (ie four distinct lines intersecting at a smooth point of X). Explicit equations of three 1-dimensional families of smooth quartic surfaces with 58 lines, and of a quartic surface with 8 singular points and 48 lines are provided.

Symmetries and equations of smooth quartic surfaces with many lines

We provide explicit equations of some smooth complex quartic surfaces with many lines, including all 10 quartics with more than 52 lines. We study the relation between linear automorphisms and some configurations of lines such as twin lines and special lines. We answer a question by Oguiso on a determinantal presentation of the Fermat quartic surface.

Curves on Heisenberg invariant quartic surfaces in projective 3-space

This paper is about the family of smooth quartic surfaces that are invariant under the Heisenberg group H_{2,2}. For a very generic X, we show that the Picard number of X is 16 and determine its Picard lattice. It turns out that a very generic X contains 320 irreducible conics which generate the Picard group of X.

Quartic surfaces with icosahedral symmetry

AbstractWe study smooth quartic surfaces in ℙ3which admit a group of projective automorphisms isomorphic to the icosahedron group.

AT MOST 64 LINES ON SMOOTH QUARTIC SURFACES (CHARACTERISTIC 2)

Let $k$ be a field of characteristic $2$. We give a geometric proof that there are no smooth quartic surfaces $S\subset \mathbb{P}_{k}^{3}$ with more than 64 lines (predating work of Degtyarev which improves this bound to 60). We also exhibit a smooth quartic containing 60 lines which thus attains the record in characteristic $2$.

Examples of rank two aCM bundles on smooth quartic surfaces in $${\mathbb {P}^{3}}$$ P 3

Examples of rank two aCM bundles on smooth quartic surfaces in $${\mathbb {P}^{3}}$$ P 3

On a smooth quartic surface containing 56 lines which is isomorphic as a K3 surface to the Fermat quartic

We give a defining equation of a complex smooth quartic surface containing 56 lines, and investigate its reductions to positive characteristics. This surface is isomorphic to the complex Fermat quartic surface, which contains only 48 lines. We give the isomorphism explicitly.

A census of zeta functions of quartic K surfaces over

We compute the complete set of candidates for the zeta function of a K$3$surface over$\mathbb{F}_{2}$consistent with the Weil and Tate conjectures, as well as the complete set of zeta functions of smooth quartic surfaces over$\mathbb{F}_{2}$. These sets differ substantially, but we do identify natural subsets which coincide. This gives some numerical evidence towards a Honda–Tate theorem for transcendental zeta functions of K$3$surfaces; such a result would refine a recent theorem of Taelman, in which one must allow an uncontrolled base field extension.

112 LINES ON SMOOTH QUARTIC SURFACES (CHARACTERISTIC 3): Table 1

Over a field |$k$| of characteristic |$3$|⁠, we prove that there are no smooth quartic surfaces |$S \subset {\mathbb {P}}^3_k$| with more than 112 lines. Moreover, the surface with 112 lines is projectively equivalent over |$\bar k$| to the Fermat quartic. As a key ingredient, we derive a characteristic-free upper bound for the number of lines met by a conic on a smooth quartic surface.

64 lines on smooth quartic surfaces

Let k be a field of characteristic other than 2,3. We prove that there are no geometrically smooth quartic surfaces in IP^3 with more than 64 lines. As a key step, we derive the sharp bound that any line meets at most 20 other lines on a smooth quartic.

Variation of Néron–Severi Ranks of Reductions of K3 Surfaces

We study the behavior of geometric Picard ranks of K3 surfaces over under reduction modulo primes. We compute these ranks for reductions of smooth quartic surfaces modulo all primes p < 216 in several representative examples and investigate the resulting statistics.

Ulrich bundles on quartic surfaces with Picard number 1

In this note, we prove that there exist stable Ulrich bundles of every even rank on a smooth quartic surface X⊂P3 with Picard number 1.

Pfaffian quartic surfaces and representations of Clifford algebras

Given a general ternary form f=f(x_1,x_2,x_3) of degree 4 over an algebraically closed field of characteristic zero, we use the geometry of K3 surfaces and van den Bergh's correspondence between representations of the generalized Clifford algebra C_f associated to f and Ulrich bundles on the surface X_f:={w^4=f(x_1,x_2,x_3)} \subseteq {P}^3 to construct a positive-dimensional family of 8-dimensional irreducible representations of C_f. The main part of our construction, which is of independent interest, uses recent work of Aprodu-Farkas on Green's Conjecture together with a result of Basili on complete intersection curves in {P}^3 to produce simple Ulrich bundles of rank 2 on a smooth quartic surface X \subseteq {P}^3 with determinant O_X(3). This implies that every smooth quartic surface in {P}^3 is the zerolocus of a linear Pfaffian, strengthening a result of Beauville-Schreyer on general quartic surfaces.

Rationality of an Enriques-Fano threefold of genus five

We prove the rationality of a non-Gorenstein Fano threefold of Fano index one and degree eight having terminal cyclic quotient singularities and Picard group . This threefold can be described as the quotient of a double covering of ramified in a smooth quartic surface by an involution fixing eight different points.

Smooth quartic surfaces with 352 conics

Up to now the maximal number of smooth conics, that can lie on a smooth quartic surface, seems not to be known. So our number 352 should be compared with 64, the maximal number of lines that can lie on a smooth quartic [S]. We construct the surfaces as Kummer surfaces of abelian surfaces with a polarization of type (1, 9). Using Saint-Donat’s technique [D] we show that they embed in IP3. In this way we only prove their existence and do, unfortunately, not find their explicit equations. So there are the following obvious questions, which we cannot answer at the moment: