Math 584 Final Exam Fall 2020: Your Final Paper And Poster
Math 584 Final Exam Fall 2020your Finalpaper And Poster Are Due Dec 17
Math 584 Final Exam Fall 2020your Finalpaper And Poster Are Due Dec 17
MATH 584 FINAL EXAM Fall 2020 Your final paper and poster are due Dec 17th describing the properties of your cubic surface and related curves including appropriate computer-generated images. Please make your title interesting and include an abstract at the beginning of the paper. Submit as pdfs and electronic files in CANVAS (i.e. pdf+ source MSword, Latex files, (as text) , ppt, etc. Include your computer codes if you use them for calculations.) The points colored in purple are to be turned in as a draft on Dec 9th for the review and input. They will be graded as midterm work and revised to be included in your final paper.
Below is the structure of the paper. The poster should summarize your work in a form that can be presented at a conference next semester (see the attached template). Start your paper with points 1 and 2 below, and then follow the other requirements (you can change the order). SURFACES 1. Title, name, abstract – (write this part at the end highlighting your results).
2. Give basic definitions of P3 ( x,y,z,w ), an algebraic variety in P3 , an irreducible variety, singular points, a dimension a variety. 3. Give the homogenous equation f of your surface V in P3 ( x,y,z,w ). What is the space classifying all degree 3 surfaces in P3 ? What is the dimension of this classifying space? Show the calculations. 4. Give definitions of Ux, Uy, Uz, Uw. . Graph your surface for w=1 , x=1, y=1, z=1 (use Surfer or any other graphing software). 5. Find singular points of V in P3 or prove there are none. Find the inflection points on your surface or prove there are none. 6. Pick a non-singular point and write the equation of a tangent space. What dimension is your variety at non-singular points? 7. Calculate Gaussian curvature of your surface V given by the equation F(x,y,z) = 0 in R3 = Uw , at least two non-singular points. Analyze the changes of the curvature on your surface? Is it always positive or negative? Is there a curve when the curvature equals always to 0? 8. Describe the ideal I( V ). Is it prime? Is your variety irreducible in P3 (justify)? 9. Describe the ring O ( V ) of regular functions on your surface. Describe the field of rational functions K ( V ). Is your V birational to P2? 10. Describe symmetries of your V , Aut( V ) – give generators or matrices if possible. You can consult these: Example: Aut (Sphere} = ( mP , rl,a id, where P is any plane passing through, the center , l is any line passing through the center , and a is any angle } is infinite, and can be described by the group of orthogonal 3 x 3 matrices O(3). 11. Find lines on your variety by solving the equations in variables s and t (you may use computers) or justify that there are none. 12. Consider curves (divisors) on V given by V intersected with a plane x=0, y=0, z=0, w= 0. Calculate genus of each curve, if possible. (The genus formula for a smooth curve on a plane is g = ( d -1)( d -2)/2 , where d is a degree of the polynomial defining the curve). 13. What can you say about the family of curves given by equations x = a (degree, irreducibility, singularities, genus, etc.)? What can you say about the family of curves given by equations y = b ? What can you say about the family of curves given by equations z = c ? What can you say about the family of curves given by equations w = d ? 14. Are there any other interesting curves that lie on your variety V (i.e. not plane sections)? For example, a twisted cubic or an elliptic curve ( g =1) may lie on your surface. 15. Define a family of (interesting) deformations of V parameterized by a in R1 . What happens to irreducibility, singularities, symmetries, lines, etc.? What happens when the parameter a goes to +infinity, - infinity? Show appropriate images. 16. Bibliography – cite all sources you have used, including our textbook and a calc book. See the example below (note: only names of Journals and title of books are in italics ): [1] A. Bremner, A. CHoudhry, M. Ulas, Constructions of diagonal quartic and sextic surfaces with infinitely many rational points. International J of Number Theory, 2014. [2] F. Catanese, G. Ceresa. Constructing Sextic Surfaces with a Given Number of Nodes. J. Pure Appl. Algebra 23 , 1-12, 1982. [3] W. Barth. Two projective surfaces with many nodes, admitting the symmetries of the icosahedron, Journal of Algebraic Geometry 5 (1): 173–186, 1996. [4] D. Jaffe, D. Ruberman. A sextic surface cannot have 66 nodes, Journal of Algebraic Geometry 6 (1): 151–168, 1997. [5] S. EndraàŸ. Surfer. A project of the Mathematisches Forschungsinstitut Oberwolfach and the Technical University Kaiserslautern (2008). [6] J. Harris. Algebraic Geometry: a first course. Springer, GTM 133, 1992. [7] G. Salmon. A treatise on the analytic geometry of three dimensions . Dublin: Hodges, Smith, & Co., 1865. [8] accessed on Nov 29, 2020. Draft 50 Final- Revit Drawing FALL 2020 Draw the floor plan below in Revit. Include all doors and windows. • Use the residential template. Fill in your name in the title block. • Wall height are all 10’-0†tall. • Please use some type of exterior wall when drawing. • Exterior walls are 6â€. • Interior walls are standard 2 x 4 construction with ½â€ drywall on each side. • Draw walls, doors and windows only. Toilets, sinks, cabinets are not required. • Extra points if you draw a roof plan and I can see it when I use the 3D icon. • You may select the sizes for the windows and doors. You can consult your book for tips on how to draw the plan. You will be graded on selection of correct walls and completeness. Submit the Revit file once it is completed. I HAVE EXTENDED THE TIME FOR YOU TO COMPLETE THIS PROJECT. DRAWING MUST BE SUBMITTED BY 11:59 ON THURSDAY, DEC 17, 2020.
Paper For Above instruction
The assignment requires a comprehensive exploration of a cubic surface in projective three-space (P3) and related algebraic and geometric properties. This includes defining the ambient projective space, understanding the nature of algebraic varieties, and analyzing specific features of the chosen surface, such as singularities, inflection points, symmetries, lines, and special curves. The paper must incorporate both theoretical calculations and computer-generated visualizations to illustrate the surface and its associated curves. Additionally, the project involves studying the deformation behavior and birational properties of the variety, culminating in an academic presentation suitable for conference purposes.
To structure the work, begin with the fundamental definitions: describe P3, algebraic varieties, irreducibility, singular points, and the dimension of varieties. Then, specify the homogeneous polynomial defining your cubic surface, discuss the classifying space of degree 3 surfaces in P3, and calculate its dimension. Proceed to define the coordinate hyperplanes Ux, Uy, Uz, Uw, and generate visualizations for the surface in various slices.
Further analysis should include identifying singular and inflection points, deriving tangent spaces at non-singular points, and calculating Gaussian curvature at multiple points to understand curvature behavior. The paper should also describe the ideal defining the surface, discuss its primality and irreducibility, and analyze the ring of regular functions along with the field of rational functions, exploring birational equivalences with projective planes.
Symmetry groups of the surface, lines lying on the surface, and specific curves—such as intersections with coordinate hyperplanes—must be investigated. Calculate the genus of these curves where applicable and analyze families of curves within the surface, assessing properties like irreducibility, singularities, and genus. The study extends to identifying other notable curves such as rational or elliptic curves embedded in the surface. Additionally, explore deformations of the surface parametrized by a real variable, observing their effect on the structure and symmetries of the surface.
Lastly, compile a comprehensive bibliography citing scholarly articles, textbooks, and online sources used during research. The entire report must be written in an academic style, fully developed with clear explanations, mathematical rigor, and visual aids, aiming to demonstrate deep understanding and original analysis of the cubic surface and its properties.