Topological Aspects of Algebraic Geometry Flashcards
Discuss degree 2 curves in R^2
What simplifications do we make to allow a more reasonable classification?
Now discuss degree 2 w.r.t. simplifications
Singular vs nonsingular
- Consider curves in RP^2 instead of R^2 - compactify
- Restrict attention to nonsingular curves
d=2
circle with complement disc U mobius band
or
empty set
pg 1-2
What is Hilbert’s 16th Problem?
Status?
Fix integer d > 0. (RP^2, RA) classify these topological pairs where RA - set of real points of nonsingular algebraic curve of degree d
Status: Unresolved for degree > 7
pg 3
From Wiki: Describe relative positions of ovals originating from a real algebraic curve.
In 1876, Harnack investigated algebraic curves in the real projective plane and found that curves of degree n could have no more than (n^2-3n+4) / 2
separate connected components. Furthermore, he showed how to construct curves that attained that upper bound, and thus that it was the best possible bound. Curves with that number of components are called M-curves.
Hilbert had investigated the M-curves of degree 6, and found that the 11 components always were grouped in a certain way. His challenge to the mathematical community now was to completely investigate the possible configurations of the components of the M-curves.
Discuss presence of pseudo-lines in classification of algebraic curves
d odd => one line
d even => no lines (only ovals)
pg 3-5
Discuss classification of algebraic curves of degree 3
psuedo-line U oval or psuedo-line
assume d odd => one line
discuss how to realize elements of classification
pg3, 7
Discuss sign w.r.t. polynomial on RP^2
Relation to pseudo-lines?
If d even, F has a well-defined sign at each point of RP^2.
With d even, an algebraic curve divides RP^2 into + half and - half. Common boundary = curve = 0 locus.
Each time you cross boundary, change the half - pseudo-line one-sided - doesn’t divide - sign doesn’t change - can’t be present in an even degree curve
Discuss unique conic through 5 points
Conic given by homogeneous poly in 3 variables of degree 2
Described by sequence of coeff.
Curves = 0 sets defined by coeff up to multiplication by nonzero constant
RP^5
Passing through a point of RP^2 defines a linear condition on coefficients - hyperplane in RP^5 – 5 hyperplanes have common point
Discuss classification of algebraic curves of degree 4
Know just ovals since degree even.
Discuss why < 5 ovals - Bezut
Discuss realizing 3, 4 ovals
pg 8
Discuss classification of algebraic curves of degree 5
Exercise…
What is Harnack thm/inequality? Proof?
Sharp bound on number of connected components of RA and statement that all values are realized.
(d-1)(d-2)/2 +1
Pf. Idea: Use auxiliary curve of degree d-2 intersect with degree d curve and find contradiction using Bezout thm.
pg 13 -15
Discuss realization of Harnack
Non-singular stb maximal or M-curve if real point set has (d-1)(d-2)/2 +1 c.c.
pg 16-17
Discuss topology of CA where A a curve in CP^2
Riemann surface of genus g = (d-1)(d-2)/2
Proof?
18-20
How does info about CA yield info about RA? Klein’s proof of Harnack?
RA = fixed points of orientation reversing involution –> complex conj.
pg 21-22
