Shuijing Crystal Li

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• Shuijing Crystal Li
• Rice University
• Mathematics Department

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An open problem
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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
• X is birational to a conic bundle over a conic
• 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?
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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.
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Lets 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.
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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?
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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.
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Remark
We can easily seen from the table above,
the structure of del pezzo surfaces of degree
dgt4 are relatively easy. For 7gtdgt5,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
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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 (xyzw) 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.

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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 -
  (xyzw)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

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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 )
  - (xyzw)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

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Rational Points on Cubic Surface on Cubic
Surface
Theorem ( Janos Kollar, 2000)
Theorem ( Unirationality of del Pezzo surface)
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From Kollars 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?
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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?
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Theorem ( A. Weil)
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Possible Unique rational point over F3
Possible Unique rational point over F7
Possible Unique rational point over F4
Possible Unique rational point over F2
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Before running to computer for help, is there any
other way?
YES! Geometry
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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
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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
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Del Pezzo Surface of degree 2
Same Question Same Approach
Different Answer
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Finally, using computer program running through
all possible coefficients.
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Theorem
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Acknowledgement
Professor Brendan Hassett Professor Robert
Hardt Professor Ron Goldman Funding Supported
by NSF