> HJG%` bjbj"x"x 3$@@4wyyyC
%
3$4ha7b3777337*w7w:,WP6aC
w304M
77W7W >F,r$33
47777
1. Name of Course: Algebraic Curves
2. Lecturer: Andras Nemethi
3. No. of Credits: 3, and no. of ECTS credits: 6
4. Semester or Time Period of the course: Autumn Semester of AY 2008-2009
5. Any other required elements of the department (eg is this course the co-requisite of another course? Is this course a pre-requisite? Etc): -
6. Course Level: advanced
7. Brief introduction to the course:
Basic principles and methods concerning algebraic geometry and plane algebraic curves are discussed. The main concepts (intersection multiplicity, degree, Bezout theorem, genus of a curve, local singularity type, Piuseux parametrizations, normalization, delta-invariant, local topological type, semigroup of the singularity, local embedded singularity link, Milnor number, local monodromy, homology of the projective singular curve) are addressed, with special emphasis on different classical problems (properties of cubics, classification of algebraic links). Many applications and examples are discussed with different connections with knot theory and affine algebraic geometry.
The course is designed for students oriented to algebraic geometry, singularity theory and/or topology.
8. The goals of the course:
The main goal of the course is to introduce students in the theory of complex affine/projective plane curves via their standard algebraic and topological invariants. We also intend to discuss different connections with knot theory, topology, and classical problems of algebraic curves.
9. The learning outcomes of the course:
The students will learn important notions and results in introductory algebraic geometry, singularity theory and topology and related invariants associated with complex projective algebraic plane (with singularities). They will gain crucial skills and knowledge in several parts of modern mathematics. Via the exercises, they will learn how to use these tools in solving specific algebraic and topological problems about these curves.
10. More detailed display of contents.
Week 1: Complex multivariable polynomials (definitions, ring structure, irreducible polynomials, ideals, prime ideals).
Week 2: Plane curves, smooth points, singular points (definitions, examples).
Week 3: Homogeneous polynomials, projective space, projective curves (examples,
affine charts).
Week 4: Local intersection multiplicity, local tangent cones, Bezuot theorem (equivalent definitions, examples, applications).
Week 5: Smooth projective cubics (group structure, Hessian, inflection points).
Week 6: Local singularity theory of plane curves (examples, Milnor number, Milnor fiber, embedded link, monodromy, Newton diagram).
Week 7: Normalization of the local singularity (Puiseux parametrization, delta-invariant, semigroup, examples, applications).
Week 8: Local topological type (embedded link) (classification results, equivalent
characterizations).
Week 9: The genus/homology of a smooth projective curve (genus formula, applications).
Week 10: The homology of singular projective urves (proof, applications).
Week 11: Divisors, lineas equivalence of divisors, the Class group (definitions, examples, the case of smooth cubics).
Week 12: Linear systems and their dimensions (applications)
Optional material: Differential forms, canonical divisor, Riemann-Roch theorem.
Books:
Brieskorn E. and Knorrer, H.: Algebraic Plane Curves.
Wall, C.T.C.: Singular points of plane curves.
Teaching format: lecture combined with classroom discussions.
11. Assessment: Written final exam within a couple of weeks after the last lecture.
Moreover, each student will give presentations about subject (theorem, interesting property) related with the course.
12. Such further items as assessment deadlines, office hours, contact details etc are at the discretion of the department or the individual.
Office hours: by appointment
Andras Nemethi
Renyi Institute of Mathematics, Budapest, nemethi@renyi.hu
&*24CWXsuyV b e n r
!op$t.3lmX*7IY
h6CJhh5CJ\
h5CJ
h>*CJ
hCJU&'CDuvR S n o B
$a$op#$Xlm''^#)*HIY^
&F
&Fh^h
hCJ
h0J1:pc;0/ =!"#$%r@rNormal$
*$1$3$5$7$A$a$+B*CJOJPJQJ^J_HaJmH sH tH@@@ Heading 1@&
&F$CJV@V Heading 2@&
&F<$OJQJCJ65DADDefault Paragraph FontVi@VTable Normal :V44
la(k@(No List:O:Numbering Symbols6U@26 HyperlinkB*ph>*POPWW-Absatz-Standardschriftart@O!@WW-Numbering SymbolsJO1JWW-Default Paragraph Font6B@B6 Body Text
xBOBBHeading
x$OJQJCJ$/@Ab$List@Or@Caption1
xx$CJ6*O*Index$$&'CDuvRSnoBop # $
Xlm
'
^
#)*HIY0800000000000000000000000000000000000000000000000000000000 0 0X0X0X0X0X0X0X0X0X0000'
A4BL$CDެp
p
yyv
v
8*urn:schemas-microsoft-com:office:smarttagsCity9*urn:schemas-microsoft-com:office:smarttagsplace\74:;B17krsc i e
n
(.
'
0
8
?
IOPWY^%
Xiru3333333333^`.77^7`.RR^R`.nn^n`.^`.^`.^`.^`. ^ `.^`^`^`^`^`^`^`^`^`c@\\ipp://172.31.12.30\Z14_314Ne02:winspoolSamsung ML-2850 Series\\ipp://172.31.12.30\Z14_314
4dXXA4PRIVo
JociUntitledddX2222222222Joci\\ipp://172.31.12.30\Z14_314
4dXXA4PRIVo
JociUntitledddX2222222222JociWP@UnknownGz Times New Roman5Symbol3&z ArialA&AlbanyArialBAhd2Ҧd2Ҧd2ҦSB
SB
!242HX)?c2Course Design TemplateCEUCEUOh+'0
8DP
\hpxCourse Design TemplateCEUNormalCEU2Microsoft Office Word@F#@X5@X5@X5SB
՜.+,0hp
Central European University'Course Design TemplateTitle
!"#$%&'()*+,-./012345689:;<=>@ABCDEFIRoot Entry F@:6KData
1Table7WordDocument3$SummaryInformation(7DocumentSummaryInformation8?CompObjq
FMicrosoft Office Word Document
MSWordDocWord.Document.89q