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Algebraic Curves
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Algebraic Curves
Fall 2003
Lecturer
Johan P. Hansen
Kontor: A3.21
Telefon: 8942 3449
E-post:
Content
Algebraic geometry is the study of polynomial equations in one or
more variables,asking such questions as: Does the system have
finitely many solutions, and if so can one find them? And if the
are infinitely many solutions, how can they be described and
manipulated? The solutions of a system of polynomial equations
form a geometric object called a variety; the corresponding
algebraic object is an ideal in a polynomial ring in several
variables. There is a close relationsship between ideals and
varieties which reveals the intimate link between algebra and
geometry. The first part of the course covers affine varieties
with Hilberts Nullstellensatz and projective geometry with Bezouts
theorem.
Depending on interest the second part of the course can cover the
theorem of Riemann-Roch, aspects of the theory elliptic curves or
dimension theory.
The algorithmic aspect of the course is based on the methods and
theories on Gröbner bases developed in Algebra 1.
The participants are supposed to perform a project (individual or
in groups). The project shall contain a written and an oral
presentation.
Students who do not intend to take a degree in Mathematics or
Statistics from the University of Aarhus, but wish to earn credits
for a 2.dels course from the Department of Mathematics, should
indicate at the beginning of the course that they wish to be
examined.
The form of examination for these students will be active
participation together with oral or written contributions.
D. Cox, J. Little and D. O'Shea, Ideals,
Varieties and Algorithms, Second Edition, Springer, UTM, 1997.
Notes
Where and when
Monday 14.15 - 16.00 in H.2.28 and
Thursday 13.15 - 15.00 in Aud. D3
Remarks
The first lecture is tuesday the 2. of september:
9.15-10.
On thursday the 11. of september all lectures at the university are
cancelled.
On monday the 6. and the 13. of october there will be no lectures.
Starting monday the 20. of october monday lectures will be in koll. G.
On monday the 17. of november lectures will be 15.15-16.00.
On thursday the 27. of november there will be no lectures.
Weekly reports and plans
Week 36: Chap. 1.1, 1.2, 1.4, (1.5) and Chap 4.1 (in part), 4.2 (in part).
Recommended exercises: Paragraph 2: 1-8. Shown that the curve with equation
x^3+y^3=1 has no rational paramatrization.
Week 37: Chap 4.2 and Chap 3.1, 3.5, 3.6. (Proof of Hilberts
Nullstellensatz).
3. Recommended exercise: 17 on p. 46 and 11 on p. 180, 7
on p. 180, 17 on p. 157.
Week 38: Chap 3.6, 4.3, 4.5, 4.6 (p.200 l. -6 to p. 201 l. -6, p.203 l. -16 to p. 204 l. 21), 5.1, 5.4 (p. 235 l. -18
to p. 236 l. 4 and p. 239 l. -14 to p. 242 l. 5).
Week 39: Chap 5.5 Recommended exercises: p. 252 exercises: 4 and 7. Chap.
8.1 and 8.2
Week 43: Group structure on an elliptic curve, see J. H. Silverman: The
Arithmetic of Eliptic Curves, p. 55-60. Zeta-function for elliptic curves, see
http://home.imf.au.dk/matjph/zeta.pdf
Week 44: The dimension of affine varieties
Week 45: The dimension of projective varieties. Elementary properties of
dimension
Week 46: Dimension and nonsingularity
Week 47:On monday the 17. of november lectures will be 15.15-16.00.
On thursday the 27. of november there will be no lectures.
Week 49:
Divisors and Riemann-Roch
Decomposition into irreducibles. Is there an algorithm for deciding if a given ideal is prime? Is there an algorithm
for deciding if a given affine variety is irreducible? Is there an algorithm for finding the minimal decomposition of a
given variety or radical ideal? Marie Jensen og Louise Pold Thomsen.
http://home.imf.au.dk/matjph/Algprojekt.pdf
Projective closure of affine varieties. Describe, implement and
apply an algorithm for
determining the smallest projective variety containing a given affine variety.
http://home.imf.au.dk/matjph/projlukning.pdf.
Allan Hansen og Thomas Koelbaek.
Mordell's theorem. The rational points on an elliptic curve is a
finetely generated abelian group. http://home.imf.au.dk/matjph/Mordell. Henrik Christensen og Michael Pedersen.
Decomposition into irreducibles. Is there an algorithm for deciding if a given ideal is prime? Is there an algorithm
for deciding if a given affine variety is irreducible? Is there an algorithm for finding the minimal decomposition of a
given variety or radical ideal? Starting ref. p. 201-206. (Marie og Louise).
Invariant theory for finite matrix groups. Does the orbit space
for a finite matrix group acting on affine space have the structure of an
affine variety? Starting ref. p. 346. (Tarik Rian and Michael Knudsen),se
http://home.imf.au.dk/matjph/invariant.ps .
Projective closure of affine varieties. Describe, implement and
apply an algorithm for
determining the smallest projective variety containing a given affine variety.
Starting ref. Chap. 8.4. (Reserved).
Intersections multiplicities defined locally. Local methods for
computing multiplicities are available. Starting ref. W. Fulton, Algebraic
Curves.
Mordell's theorem. The rational points on an elliptic curve is a
finetely generated abelian group. Starting ref. J. H. Silverman and J. Tate:
Rational Points on Elliptic Curves.
Points of finite order on elliptic curves.
Starting ref. Silver and Tate: Rational points on elliptic curves, Chap. II
Blowing-up a variety (reserved).
Starting ref. exercises. 14-17 on p. 494-495.
Elliptiske kurver og Weierstrass p-funktioner, starting ref. Silverman: The Arithmetic of Elliptic Curves, p. 150-165
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