scholarly journals Surfaces with canonical map of maximum degree

2021 ◽  
Vol 31 (1) ◽  
pp. 127-135
Author(s):  
Carlos Rito
Keyword(s):  
Finite Field ◽  
Projective Plane ◽  
Maximum Degree ◽  
Irregular Surface ◽  
Regular Surface ◽  
Content Type ◽  
Canonical Map

We use the Borisov-Keum equations of a fake projective plane and the Borisov-Yeung equations of the Cartwright-Steger surface to show the existence of a regular surface with canonical map of degree 36 and of an irregular surface with canonical map of degree 27. As a by-product, we get equations (over a finite field) for the Z / 3 \mathbb {Z}/3 -invariant fibres of the Albanese fibration of the Cartwright-Steger surface and show that they are smooth.

scholarly journals Cubic Arcs in the Projective Plane Over a Finite Field of Order Twenty Three

2019 ◽  
Vol 54 (6) ◽  
Author(s):  
Najm A.M. Al-Seraji ◽  
Asraa A. Monshed
Keyword(s):  
Finite Field ◽  
Projective Plane ◽  
Coding Theory ◽  
Finite Projective Plane ◽  
Cubic Curves ◽  
The Subject

In this research we are interested in finding all the different cubic curves over a finite projective plane of order twenty-three, learning which of them is complete or not, constructing the stabilizer groups of the cubics in, studying the properties of these groups, and, finally, introducing the relation between the subject of coding theory and the projective plane of order twenty three.

Magic squares of squares over a finite field

2021 ◽  
pp. 111-122
Author(s):  
Stewart Hengeveld ◽  
Giancarlo Labruna ◽  
Aihua Li
Keyword(s):  
Finite Field ◽  
Prime Numbers ◽  
Similar Question ◽  
Magic Squares ◽  
Content Type ◽  
Magic Square ◽  
Twin Primes ◽  
Quadratic Residues ◽  
Perfect Squares ◽  
Open Question

A magic square M M over an integral domain D D is a 3 × 3 3\times 3 matrix with entries from D D such that the elements from each row, column, and diagonal add to the same sum. If all the entries in M M are perfect squares in D D , we call M M a magic square of squares over D D . In 1984, Martin LaBar raised an open question: “Is there a magic square of squares over the ring Z \mathbb {Z} of the integers which has all the nine entries distinct?” We approach to answering a similar question when D D is a finite field. We claim that for any odd prime p p , a magic square over Z p \mathbb Z_p can only hold an odd number of distinct entries. Corresponding to LaBar’s question, we show that there are infinitely many prime numbers p p such that, over Z p \mathbb Z_p , magic squares of squares with nine distinct elements exist. In addition, if p ≡ 1 ( mod 120 ) p\equiv 1\pmod {120} , there exist magic squares of squares over Z p \mathbb Z_p that have exactly 3, 5, 7, or 9 distinct entries respectively. We construct magic squares of squares using triples of consecutive quadratic residues derived from twin primes.

Direct Product Decompositions of Elation Groups

1977 ◽  
Vol 20 (2) ◽  
pp. 173-182
Author(s):  
Julia M. Nowlin Brown
Keyword(s):  
Finite Field ◽  
Projective Plane ◽  
Direct Product ◽  
Collineation Group

Let G be a collineation group of a projective plane π. Let E be the subgroup generated by all elations in G. In the case that π is finite and G fixes no point or line, F. Piper [6; 7] has proved that if G contains certain combinations of perspectivities, then E is isomorphic to for some finite field g.

scholarly journals Topological Realizations of Line Arrangements

2017 ◽  
Vol 2019 (8) ◽  
pp. 2295-2331
Author(s):  
Daniel Ruberman ◽  
Laura Starkston
Keyword(s):  
Finite Field ◽  
Projective Plane ◽  
Contact Structure ◽  
Complex Projective Plane ◽  
Real Projective Plane ◽  
Incidence Relation ◽  
Combinatorial Configuration ◽  
Important Variation ◽  
Graph Manifolds ◽  
Complex Projective

Abstract A venerable problem in combinatorics and geometry asks whether a given incidence relation may be realized by a configuration of points and lines. The classic version of this would ask for lines in a projective plane over a field. An important variation allows for pseudolines: embedded circles (isotopic to $\mathbb R\rm{P}^1$) in the real projective plane. In this article we investigate whether a configuration is realized by a collection of 2-spheres embedded, in symplectic, smooth, and topological categories, in the complex projective plane. We find obstructions to the existence of topologically locally flat spheres realizing a configuration, and show for instance that the combinatorial configuration corresponding to the projective plane over any finite field is not realized. Such obstructions are used to show that a particular contact structure on certain graph manifolds is not (strongly) symplectically fillable. We also show that a configuration of real pseudolines can be complexified to give a configuration of smooth, indeed symplectically embedded, 2-spheres.

scholarly journals Counting the number of trigonal curves of genus 5 over finite fields

2020 ◽  
Vol 208 (1) ◽  
pp. 31-48
Author(s):  
Thomas Wennink
Keyword(s):  
Finite Field ◽  
Projective Plane ◽  
Euler Characteristic ◽  
Finite Fields ◽  
Moduli Space ◽  
Plane Curves ◽  
Sieve Method ◽  
Main Application ◽  
Trigonal Curves

AbstractThe trigonal curves of genus 5 can be represented by projective plane quintics that have one singularity of delta invariant one. Combining this with a partial sieve method for plane curves we count the number of such curves over any finite field. The main application is that this gives the motivic Euler characteristic of the moduli space of trigonal curves of genus 5.

Radical orbital decompression for severe dysthyroid exophthalmos

1982 ◽  
Vol 56 (2) ◽  
pp. 260-266 ◽  
Author(s):  
Joseph C. Maroon ◽  
John S. Kennerdell
Keyword(s):  
Surgical Technique ◽  
Surgical Procedure ◽  
Maximum Degree ◽  
Skin Incision ◽  
Orbital Decompression ◽  
Content Type ◽  
Decompression Procedure ◽  
Unilateral Proptosis ◽  
Lateral Canthotomy ◽  
Fine Print

✓ A surgical technique is described for radical decompression of the orbit for dysthyroid exophthalmopathy. The operation should be considered in a patient with proptosis greater than 30 mm bilaterally or one with unilateral proptosis of 10 mm or more greater than the opposite eye. Such exophthalmos is frequently associated with corneal exposure and ulceration, extreme cosmetic disfigurement, and optic neuropathy. The surgical procedure is performed through a 35-mm lateral skin incision and a lateral canthotomy with subconjunctival dissection. All four walls of the orbit are partially removed. This panorbital decompression procedure has been performed on five patients, with reduction of preoperative proptosis by as much as 17 mm. Complications were minimal. A review of the effectiveness of other orbital decompressive procedures is presented. It appears that the four-wall decompressive procedure offers the maximum degree of orbital reduction.

scholarly journals Spanning the isogeny class of a power of an elliptic curve

2021 ◽  
Vol 91 (333) ◽  
pp. 401-449
Author(s):  
Markus Kirschmer ◽  
Fabien Narbonne ◽  
Christophe Ritzenthaler ◽  
Damien Robert
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Elliptic Curve ◽  
Finite Fields ◽  
Modular Forms ◽  
Null Point ◽  
Siegel Modular Forms ◽  
Content Type ◽  
Careful Choice ◽  
Isomorphism Classes

Let E E be an ordinary elliptic curve over a finite field and g g be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class of E g E^g . The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null point. We then use these results to give an algebraic computation of Serre’s obstruction for principally polarized abelian threefolds isogenous to E 3 E^3 and of the Igusa modular form in dimension 4 4 . We illustrate our algorithms with examples of curves with many rational points over finite fields.

scholarly journals Some Applications of Coding Theory in the Projective Plane of Order Four

2019 ◽  
Vol 30 (1) ◽  
pp. 152
Author(s):  
Najm A. M. AL-Seraji ◽  
Hamza L. M. Ajaj
Keyword(s):  
Finite Field ◽  
Projective Plane ◽  
Coding Theory ◽  
Minimum Distance ◽  
Incidence Matrix ◽  
Maximum Value ◽  
Order Four ◽  
The Relationship

The main aim of this research is to introduce the relationship between the topic of coding theory and the projective plane of order four. The maximum value of size code M over the finite field of order four and an incidence matrix with the parameters, n (length of code), d (minimum distance of code) and e (error-correcting of code) have been constructed. Some examples and theorems have been given.

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