An arithmetic count of the lines on a smooth cubic surface

We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field $k$ , generalizing the counts that over ${\mathbf {C}}$ there are $27$ lines, and over ${\mathbf {R}}$ the number of hyperbolic lines minus the number of elliptic lines is $3$ . In general, the lines are defined over a field extension $L$ and have an associated arithmetic type $\alpha$ in $L^*/(L^*)^2$ . There is an equality in the Grothendieck–Witt group $\operatorname {GW}(k)$ of $k$ , \[ \sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle \alpha \rangle = 15 \cdot \langle 1 \rangle + 12 \cdot \langle -1 \rangle, \] where $\operatorname {Tr}_{L/k}$ denotes the trace $\operatorname {GW}(L) \to \operatorname {GW}(k)$ . Taking the rank and signature recovers the results over ${\mathbf {C}}$ and ${\mathbf {R}}$ . To do this, we develop an elementary theory of the Euler number in $\mathbf {A}^1$ -homotopy theory for algebraic vector bundles. We expect that further arithmetic counts generalizing enumerative results in complex and real algebraic geometry can be obtained with similar methods.

Cohomology of the universal smooth cubic surface

We compute the rational cohomology of the universal family of smooth cubic surfaces using Vassiliev’s method of simplicial resolution. Modulo embedding, the universal family has cohomology isomorphic to that of $\mathbb{P}^2$. A consequence of our theorem is that over the finite field $\mathbb{F}_q$, away from finitely many characteristics, the average number of points on a smooth cubic surface is q2+q + 1.

38406501359372282063949 and All That: Monodromy of Fano Problems

A Fano problem is an enumerative problem of counting $r$-dimensional linear subspaces on a complete intersection in ${\mathbb{P}}^n$ over a field of arbitrary characteristic, whenever the corresponding Fano scheme is finite. A classical example is enumerating lines on a cubic surface. We study the monodromy of finite Fano schemes $F_{r}(X)$ as the complete intersection $X$ varies. We prove that the monodromy group is either symmetric or alternating in most cases. In the exceptional cases, the monodromy group is one of the Weyl groups $W(E_6)$ or $W(D_k)$.

Automorphisms of cubic surfaces without points

We classify finite groups acting by birational transformations of a nontrivial Severi–Brauer surface over a field of characteristic zero that are not conjugate to subgroups of the automorphism group. Also, we show that the automorphism group of a smooth cubic surface over a field [Formula: see text] of characteristic zero that has no [Formula: see text]-points is abelian, and find a sharp bound for the Jordan constants of birational automorphism groups of such cubic surfaces.

APPLICATION OF MULTI-FACETED FUNCTION MODEL BASED ON VONDRAK FILTER OPTIMIZATION IN UAV AERIAL IMAGE HEIGHT CORRECTION

With the rapid development of drone technology and digital camera technology, the method of obtaining high-precision coordinates based on UAV aerial photogrammetry technology is popular. The plane coordinate accuracy of the aerial image of the drone has been able to meet the needs of practical applications, but the elevation accuracy is generally low. Aiming at the low elevation accuracy of UAV aerial photogrammetry, a multi-face function fitting method based on Vondrak filter optimization was proposed. The improved fitting model was used to obtain the elevation correction value of the aerial image, thereby obtaining high-precision image elevation data. In this paper, based on the traditional multi-face function fitting method, some known points were used to model and find the difference between the measured elevation value and the measured elevation. The Vondrak filter was used to smooth the fitting result. Finally, a small number of known elevation points were used for checking, so that the obtained elevation was compared with the actual elevation. The experimental comparison showed that the improved multi-face function fitting method used Vondrak filter was improved by 34.76% compared with the quadric surface fitting, and improved by 14.48% compared with the optimized cubic surface fitting method. Research shows that the multi-faceted function method based on Vondrak filtering is superior to the traditional elevation correction method. The experiment verifies the effectiveness and feasibility of the improved method, and provides some reference value for the research of aerial image elevation correction model.

On space maximal curves

Any maximal curve X is equipped with an intrinsic embedding π: X → Pr which reveal outstanding properties of the curve. By dealing with the contact divisors of the curve π(X) and tangent lines, in this paper we investigate the first positive element that the Weierstrass semigroup at rational points can have whenever r = 3 and π(X) is contained in a cubic surface.

Digital Image Correlation Analysis of Displacement Based on Corrected Three Surface Fitting Algorithm

SFA (Surface Fitting Algorithm) for continuous displacement is an important method for digital image correlation with antinoise ability and computational efficiency advantages in practical applications. In order to improve the algorithm accuracy and expand its application range, this paper tries to improve the SFA and studies the modified cubic surface fitting algorithm CTSFA (Corrected Three Surface Fitting Algorithm), which is suitable for solving the initial value of continuous displacement. Bilinear interpolation and adjacent interpolation are used to analyze the gray level at any integer-pixel position in the displacement matrix and the weight coefficient is given. The distance-weighted method is used to approximate the true initial displacement value of the continuum, and the algorithm suitable for digital image processing is extended to the continuum displacement solution. The cubic surface expression of the CTSFA programmatic application is solved by the least squares method, and the correlation coefficient of the power basis function is calculated. In the computer simulation of speckle test, the comparison between CTSFA and SFA on the calculation results of linear and nonlinear displacement fields shows that the calculated amount of CTSFA is basically the same as that of SFA, but the calculation accuracy is doubled. The study of analysing the Brazilian splitting test using CTSFA and SFA reveals that CTSFA is better than SFA in observing the development of cracks.

Modelado molecular de un ánodo de carbón activado como soporte de platino por medio de cálculos DFT

In this paper we present the molecular modeling of an activated carbon anode as platinum support of a proton exchange membrane fuel cell (PEMFC), in which platinum (Pt) interacts as a catalyst supported in activated carbon (CA) with molecular hydrogen (H2) in order to break it or ionize it. The adsorption between CA and Pt is analyzed as an atom or as a minimum FCC(face-centered cubic) surface structure, to observe if there is any difference between activated and non-activated carbon in the Pt-H2 reaction. Using quantum calculations of the Density Functional Theory (DFT), the adsorption and the effect of activated carbon on platinum were investigated, using different carbon and platinum structures. Our results show that the activated carbons provide a stable support for the platinum catalysts, taking into account that the CA presents a well of lower potential energy to adsorb the platinum (physisorption), whereas an activated carbon has a potential energy well major (chemisorption).