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Examples of Cubic Surfaces

VIEWS: 8 PAGES: 26

									Rational Points on del Pezzo
Surfaces of degree 1 and 2

        Shuijing Crystal Li
         Rice University
     Mathematics Department


                               1
What is del Pezzo surface?
• Definition
• Geometric structure of del Pezzo surface


Examples of del Pezzo surface
• How could we see/understand it?
• An overview of Cubic Surfaces


Rational Points on del Pezzo surface
• Rational points on cubic surface
• Rational Points on del Pezzo surface of degree 1 and 2


                                                       2
    An open problem



z               y

         x




                      3
            Algebra
Geometry
            Solving
Variety
           equations

                       4
5
Q: Among numerous algebraic varieties, why do we care
          about “ del Pezzo surfaces”?


Theorem ( Iskovskikh)
   Given a rational variety X of dimension 2 over perfect
field k, at least one of the following happens:
    a) X is birational to a conic bundle over a conic
    b) X is k-birational to a del Pezzo surface.



  Hence, we understand the del Pezzo surfaces, we understand
    almost all the rational surfaces. How wonderful is that?



                                                               6
Our special algebraic variety : del Pezzo surfaces
Definition




Remark




Algebraic Point of View:

         From now on, we can then think of the del Pezzo surfaces as subsets
    of projective space given as the zero locus of some homogeneous polynomials.
                                                                                   7
Let’s jump to geometry




  Many interesting arithmetic questions are connected with the class of del
 Pezzo surfaces, as such surfaces are geometrically rational (i.e. rational over the
 complex field).
   It is especially interesting to look at problems concerning the question about
 the existence of k-rational points, where k is a non-closed field.
                                                                                       8
   After learning about the blow-up and using it to construct del Pezzo
 surfaces, we will turn to study their geometric structure.

  Theorem(Yu.I.Manin) Classification of del Pezzo surfaces
Let X be a del Pezzo surface of degree d.
(a) 1 ≤ d ≤ 9.
(b) (Classification) If the base field is separably closed, either:
       X is isomorphic to the blow-up of a projective plane at k = 9 − d points, or
       X is isomorphic to P1xP1 (and d = 8)
   In particular, if d = 9, then X is isomorphic to the projective plane P2 (the blow-up of
     P2 at no points).
(c) (Converse) If X is the blow-up of a projective plane at k = 9 − d points in generic
     position (no three points collinear, no six on a conic, no eight of them on a cubic
     having a node at one of them), then X is a del Pezzo surface.
     Another way to study the geometry of an algebraic surface is to look at the
curves that lie on the surface. Since our del Pezzo surfaces can be embedded into
projective space, we can study whether any “lines” in projective space are completely
contained in our surface.

           Question: Are there any “lines” on del Pezzo surfaces?
                                                                                              9
  Amazingly, the answer is YES, and there are finitely many of them! How
many? How are they configured?
 From now on, “lines” means projective lines, and will be replaced by
exceptional curves or (-1)-curves.

  Theorem Every del Pezzo surface has only finitely many exceptional curves,
 and their structure is independent of the location of the points blown up,
 provided that they are general.




                                                                               10
Remark
      We can easily seen from the table above, the structure of del pezzo
   surfaces of degree d>=4 are relatively easy. For 7>=d>=5,all Del Pezzo
   surfaces of the same degree are isomorphic.

 Back to Algebra and answer a previous question:

Theorem (Yu.I.Manin,1986) The defining equation of del Pezzo surface




                                                                            11
                      An overview of Cubic Surfaces
Cubic Surfaces
     A cubic surface is the vanishing set of a homogenous polynomial of
    degree 3 in P3, i.e. it consists of all (x:y:z:w) in P3 with:
           a0x3 + a1x2y + a2x2z + ... a18z2w + a19w3 = 0
History
•       In the 19th century, mathematicians started to study the structure of such
    vanishing sets of polynomials of different degrees in P3, called algebraic surfaces. It
    turned out, that each generic cubic surface contains 27 straight lines.

•       From this starting point, a lot of mathematicians have studied cubic surfaces
    and the structure of the 27 lines upon it.

•        In 1861, Clebsch showed, that the defining equation of a cubic surface can be
    put, in a unique way, in the so called pentahedral form, which allows us to calculate
    the equations of the 27 lines directly from the equation.


                                                                                          12
         Examples of Cubic Surfaces


                                         The Clebsch Diagonal Surface
                                    The Clebsch Diagonal Surface is one of the most
                                 famous cubic surfaces because of its symmetry and the
                                 fact that it's the only one with ten Eckardt Points.

                                    Defining equation 0= x3 + y3 + z3 + w3 - (x+y+z+w)3

                                   From the picture below the surface , we can see that
                                 there are ten Eckardt Points (points, where three lines
                                 meet in a point).




10 Eckardt points and 27 lines                                                      13
  Examples of Cubic Surfaces


                                The Cayley Cubic
                  The Cayley Cubic Surface contains four double
                  points (which is the maximum number for any
                  cubic surface).
                     Defining Equation is 4(x3 + y3 + z3 + w3 ) -
                  (x+y+z+w)3 . From the picture below the surface,
                  one sees immediately, that there are four
                  double points on the surface (each one
                  corresponds to a set of three points on a line in
                  the plane).




4 double points                                                  14
          Rational Points on Cubic Surface on
         Cubic Surface

Theorem ( Jano’s Kollar, 2000)




Theorem ( Unirationality of del Pezzo surface)




                                                 15
   From Kollar’s theorem, we discover that the unirationality of an algebraic
   surface is closely related to the existence of k-rational point.


Q : Does there exist a cubic hypersurface with unique k-rational point?




                                                                                16
                    Del Pezzo Surface of degree 1




Q : Does there really exist a del Pezzo surface of degree 1 with unique
                k-rational point over some local field?

                                         Algebraic Way: Running through all possible
                                             coefficients to find unique solution




                  Geometric Way:
   Using Weil’s Theorem and the trace information
                                                                                       17
Theorem ( A. Weil)




                     18
 Possible Unique
rational point over
         F3


   Possible Unique
    rational point
       over F7


   Possible Unique
  rational point over
           F4



     Possible Unique
      rational point
         over F2
                      19
Before running to computer for help, is there any other way?



                       YES! Geometry



                                                               20
Finally, using computer program running through all possible coefficients, then I
  found: there is no degree 1 del Pezzo surface with unique rational point over
                                  any local field

                                                                                    21
Theorem

     Let X be smooth del Pezzo surfaces of degree 1 defined
as above, then X has at least 3 rational points over any finite
field




                                                                  22
 Del Pezzo Surface of degree 2




Same Question      Same Approach



Different Answer

                                   23
Finally, using computer program running through all possible coefficients.

                                                                             24
Theorem




          25
Acknowledgement


Professor Brendan Hassett

Professor Robert Hardt

Professor Ron Goldman

Funding: Supported by NSF




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