Quartic Surface Research Articles

On an Enriques Surface Associated With a Quartic Hessian Surface

AbstractLet $Y$ 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 $Y$. 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 surface is calculated explicitly.

Smooth models of singular $K3$-surfaces

We show that the classical Fermat quartic has exactly three smooth spatial models. As a generalization, we give a classification of smooth spatial (as well as some other) models of singular K3 -surfaces of small discriminant. As a by-product, we observe a correlation (up to a certain limit) between the discriminant of a singular K3 -surface and the number of lines in its models. We also construct a K3 -quartic surface with 52 lines and singular points, as well as a few other examples with many lines or models.

A Numerical Transcendental Method in Algebraic Geometry: Computation of Picard Groups and Related Invariants

Based on high precision computation of periods and lattice reduction techniques, we compute the Picard group of smooth surfaces. We also study the lattice reduction technique that is employed in order to quantify the possibility of numerical error in terms of an intrinsic measure of complexity of each surface. The method applies more generally to the computation of the lattice generated by Hodge cycles of middle dimension on smooth projective hypersurfaces. We demonstrate the method by a systematic study of thousands of quartic surfaces (K3s) defined by sparse polynomials. As an application, we count the number of rational curves of a given degree lying on each surface. For quartic surfaces we also compute the endomorphism ring of their transcendental lattice.

Automorphisms of cubic surfaces in positive characteristic

We classify all possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic. As an intermediate step we also classify automorphism groups of quartic del Pezzo surfaces. We show that the moduli space of smooth cubic surfaces is rational in every characteristic, determine the dimensions of the strata admitting each possible isomorphism class of automorphism group, and find explicit normal forms in each case. Finally, we completely characterize when a smooth cubic surface in positive characteristic, together with a group action, can be lifted to characteristic zero.

On binary cubic and quartic forms

In this paper we determine the group of rational automorphisms of binary cubic and quartic forms with integer coefficients and non-zero discriminant in terms of certain quadratic covariants of cubic and quartic forms. This allows one to give precise asymptotic formulae for the number of integers in an interval representable by a binary cubic or quartic form and extends work of Hooley. Further, we give the field of definition of lines contained in certain cubic and quartic surfaces related to binary cubic and quartic forms.

Quartic unexpected curves and surfaces

Our research is motivated by recent work of Cook II, Harbourne, Migliore, and Nagel on configurations of points in the projective plane with properties that are unexpected from the point of view of the postulation theory. In this note, we revisit the basic configuration of nine points appearing in work of Di Gennaro/Ilardi/Vallès and Harbourne, and we exhibit some additional new properties of this configuration. We then pass to projective three-space mathbb {P}^3 and exhibit a surface with unexpected postulation properties there. Such higher dimensional phenomena have not been observed so far.

Wigner Distribution by Adiabatic Switching in Normal Mode or Cartesian Coordinates and Molecular Applications.

We recently presented a simple, classical trajectory-based method for generating the Wigner phase space density using classical trajectories evolving under an adiabatically switched potential. The adiabatically switched Wigner (ASW) distribution is an approximation to the exact Wigner function, which was found to be highly accurate on model systems. In this paper we discuss the implementation of the ASW procedure to polyatomic molecules both in normal mode coordinates and in Cartesian coordinates. We present its application to a six-degree-of-freedom model based on an ab initio quartic potential energy surface developed for formaldehyde in the normal mode representation and for butyne in Cartesian coordinates using the CHARMM force field. Comparisons of equilibrium properties against accurate quantum mechanical results indicate that the ASW is reliable and highly accurate over a wide temperature range in both the coordinate systems. Further, the ASW density is invariant under classical evolution, thus it is ideally suited to quasiclassical trajectory simulations. We also describe a very simple ASW-based procedure for obtaining complex-valued quasiclassical time correlation functions and vibrational spectra.

Counting rational points on quartic del Pezzo surfaces with a rational conic

Upper and lower bounds, of the expected order of magnitude, are obtained for the number of rational points of bounded height on any quartic del Pezzo surface over {mathbb {Q}} that contains a conic defined over mathbb {Q}.

An interesting quartic surface, everywhere locally solvable, with cubic point but no global point

An interesting quartic surface, everywhere locally solvable, with cubic point but no global point

The minimal resolution conjecture on a general quartic surface in [formula omitted

Mustaţă has given a conjecture for the graded Betti numbers in the minimal free resolution of the ideal of a general set of points on an irreducible projective algebraic variety. For surfaces in P3 this conjecture has been proven for points on quadric surfaces and on general cubic surfaces. In the latter case, Gorenstein liaison was the main tool. Here we prove the conjecture for general quartic surfaces. Gorenstein liaison continues to be a central tool, but to prove the existence of our links we make use of certain dimension computations. We also discuss the higher degree case, but now the dimension count does not force the existence of our links.

Absolute IR vibrational band intensities of hexafluoroacetone: Comparison of experiment and anharmonic ab initio calculation using the second-order operator canonical Van Vleck perturbation theory

Hexafluoroacetone (HFA, OC(CF3)2) is widely utilized in organic synthesis, chemistry of heat resistant polymers and biochemistry. In this work, infrared (IR) spectra of gaseous HFA were recorded in the range 2000–340 cm−1 with a resolution of 1 cm−1. For the first time, experimental absolute IR intensities of fourteen fundamental bands of HFA were measured and compared with intensities predicted by anharmonic ab initio calculations. Theoretical molecular structures, semi-diagonal quartic potential energy surfaces (PES), and cubic surfaces of dipole moment (DM) components of HFA were calculated using MP2/cc-pVTZ ab initio quantum-mechanical model. In addition, harmonic frequencies were also calculated at MP2/cc-pVQZ level of theory and used for refining the PES. The numerical-analytic implementation of the second-order operator canonical Van Vleck perturbation theory (CVPT2) was employed for predicting anharmonic IR spectra of HFA. The deviation from observations of predicted IR intensities for ten fundamentals (ν1, ν2, ν8, ν13–ν21) was less than 25% on average. The significant difference between ab initio harmonic intensities calculated with MP2/cc-pVTZ and MP2/cc-pVQZ models for some normal modes indicates that an inaccuracy of the quantum-mechanical model is probably the major source of errors in theoretical intensities. An interpretation of the IR experimental spectra and associated literature data for HFA is accomplished; the ν4, ν7, ν10 and ν15 fundamentals were reassigned. Three predicted Fermi resonances were identified in experimental spectra, ν3/ν5 + ν8, ν17/ν10 + ν19 and ν20/ν9 + ν23. An empirical fit of harmonic frequencies within anharmonic model assisted in a reliable identification of a number of overtones and binary combination tones. The fitted harmonic frequencies were compared with MP2/cc-pVQZ and a simplified complete basis set (CBS) extrapolation. It was found that MP2/cc-pVQZ frequencies were more reliable. The efficiency and convenience of the theoretical model employed and associated techniques for interpretation of complex spectra of HFA are demonstrated.

Vibrational spectroscopy of tolane; Coriolis coupling between Raman-active modes of g symmetry

ABSTRACTVibrational spectroscopy of tolane (diphenylacetylene), which has 66 normal modes, has been advanced. Anharmonic wavenumber predictions were made with the quartic potential energy surface obtained with B3LYP/cc-pVTZ model and the second-order perturbation theory (VPT2). Infrared (IR) intensity and Raman activities were computed at the harmonic level. The IR spectrum of the crystal and Raman spectra of the liquid and the crystal tolane were newly recorded. The lingering problem of an excess of polarised Raman bands at wavenumbers appropriate for fundamentals, other than ag modes, has now been attributed to Coriolis coupling within modes of g symmetry species. Consequently, D2h point symmetry group has been confirmed for a planar tolane molecule. Assignments for almost all fundamentals of tolane are now secure. The assignment for ν32 remains questionable. Remaining unassigned fundamentals are: ν34 and ν35, which, as au symmetry species, are IR- and Raman-inactive transitions, and ν59(b2u), which is predicted to have a very low wavenumber.

Regular pairs of quadratic forms on odd-dimensional spaces in characteristic 2

We describe a normal form for a smooth intersection of two quadrics in even-dimensional projective space over an arbitrary field of characteristic 2. We use this to obtain a description of the automorphism group of such a variety. As an application, we show that every quartic del Pezzo surface over a perfect field of characteristic 2 has a canonical rational point and, thus, is unirational.

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.

On Models and Visualizations of Some Special Quartic Surfaces

On Models and Visualizations of Some Special Quartic Surfaces

Quartic surfaces with icosahedral symmetry

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

Nonadiabatic Dynamics via the Symmetrical Quasi-Classical Method in the Presence of Anharmonicity.

The symmetrical quasi-classical (SQC) method recently proposed by Miller and Cotton allows one to simulate nonadiabatic dynamics based on an algorithm with classical-like scaling with respect to system size. This is made possible by casting the electronic degrees of freedom in terms of mapping variables that can be propagated in a classical-like manner. While SQC was shown to be rather accurate when applied to benchmark models with harmonic electronic potential energy surfaces, it was also found to become inaccurate and to suffer numerical instabilities when applied to anharmonic systems. In this paper, we propose an extended SQC (E-SQC) methodology for overcoming those discrepancies by describing the anharmonic nuclear modes, which are coupled to the electronic degrees of freedom, in terms of classical-like mapping variables. The accuracy of E-SQC relative to standard SQC is demonstrated on benchmark models with quartic and Morse potential energy surfaces.

The Picard group of various families of $(\mathbb {Z}/2\mathbb {Z})^{4}$-invariant quartic K3 surfaces

The subject of this paper is the study of various families of quartic K3 surfaces which are invariant under a certain $(\mathbb{Z}/2\mathbb{Z})^{4}$ action. In particular, we describe families whose general member contains $8,16,24$ or $32$ lines as well as the $320$ conics found by Eklund (some of which degenerate into the mentioned lines). The second half of this paper is dedicated to finding the Picard group of a general member of each of these families, and describing it as a lattice. It turns out that for each family the Picard group of a very general surface is generated by the lines and conics lying on said surface.

New surfaces with $K^2 = 7$ and $p_g = q \leq 2$

We construct smooth minimal complex surfaces of general type with $K^2=7$ and: $p_g=q=2,$ Albanese map of degree $2$ onto a $(1,2)$-polarized abelian surface; $p_g=q=1$ as a double cover of a quartic Kummer surface; $p_g=q=0$ as a double cover of a numerical Campedelli surface with $5$ nodes.

Performance of different density functionals for the calculation of vibrational frequencies with vibrational coupled cluster method in bosonic representation

An accurate description of anharmonic vibrational frequencies of polyatomic molecules is a challenging task. It requires an ab initio method to solve the vibrational Schrodinger equation along with extensive electronic structure calculations to generate the quartic potential energy surface (PES) in mass-weighted normal coordinates. The computation of such quartic PES is very expensive. Even for a medium-size molecule, highly accurate ab initio methods like CCSD, CCSD(T) become formidable. The DFT stands as valuable alternative in this case. In this work, we investigate the performances of several commonly used density functionals, namely, B3LYP, BLYP, B3LYPD, M06, M062x, PBE1PBE, B3P86, LC- $$\omega $$ PBE, X3LYP, B3PW91, and B97D for the evaluation of anharmonic vibrational frequencies of semi-rigid molecules. The quality of the results is assessed by the comparison with experimental values. To this end, we used a set of 19 molecules of various sizes (4–9 atoms). The vibrational coupled cluster method (VCCM) in bosonic representation is used to solve the vibrational structure problem. The hybrid functionals B3P86, B3LYP, B3PW91, PBE1PBE, and X3LYP found to give more accurate result of the fundamental frequencies than the other functionals. Our results show that the error in the BLYP and B97D calculation is due to the inadequate description of the harmonic force field. For the LC- $$\omega $$ PBE, M06, and M062x, the anharmonic force constants leads to the error. It is found that the comparative performances of the DFT functionals with VCCM are consistent with the second-order vibrational perturbation theory.