What are...prevarieties?
HTML-код
- Опубликовано: 5 окт 2024
- Goal.
Explaining basic concepts in the intersection of geometry and algebra in an intuitive way.
This time.
What are...prevarieties? Or: Let’s get more abstract!
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
In this course I try to cover my favorite topics in algebraic geometry, from classical ideas such as algebraic varieties, to modern ideas such as schemes, to really modern ideas such as tropical varieties. I give a very biased collection of topics, and not nearly all that can be said will be addressed. Sorry for that.
Slides.
www.dtubbenhaue...
Website with exercises.
www.dtubbenhaue...
Thumbnail.
Picture from the first video slides.
Classical algebraic geometry.
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...)
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
Modern algebraic geometry.
en.wikipedia.o...
en.wikipedia.o...)
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
Modern algebraic geometry version 2.
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
Applications of (algebraic) geometry.
math.stackexch...
Pictures used.
Picture from www.dtubbenhau...
en.wikipedia.o...
Another picture from www.dtubbenhau...
ichef.bbci.co....
Picture from agag-gathmann....
Again en.wikipedia.o...
Some books I am using (I sometimes steal some pictures from there).
agag-gathmann....
www.cambridge....
bertini.nd.edu...
mathoverflow.n...
Computer talk.
magma.maths.us...
reference.wolf...
#algebraicgeometry
#geometry
#mathematics
I’m not sure I followed how we can make a finite open cover with affine varieties.
From my understanding an open cover is a family of open sets. Affine varieties are exactly the closed sets in the Zariski topology (is that correct?)
So a finite family of affine varieties is a family of closed sets. I understand that sets can be simultaneously open and closed…. I’m just not sure I’m following what space the open cover is coming from.
Good question.
For an affine variety itself, it is open with respect to itself. So its covered by itself. (You are correct that such a variety is closed with respect to the parent space, but we can take it with respect to itself.)
In general, you want to use the so-called distinguished open subset. These work as follows. Take a polynomial function f on the variety, and take the set of elements that do not vanish f(v) not zero. These sets are open and “very large”.
Does that make sense?
@@VisualMath yes, that does make sense. I look forward to the next video! Thanks!
@@geekoutnerd7882 Welcome ☺