![]() Throughout, let $X$ be variety over a field $k$ and let $\mathcal$, then we know $V$, and vice-versa (so this definition of a linear system is equivalent to that of user42912). However, if you add nice properties to $X$ (reduced, proper, separated etc) you get at least in the complex case a space that is categorically equivalent to a complex variety (see Serres GAGA theorems).As Relapsarian mentioned in the comments, this is all discussed in Chapter II.7 of Hartshorne, but I'll try to strip away some of the language of schemes in my answer (although I assume the reader knows what a line bundle is, I provide in translation to the language of Fulton's book in the edit below). ![]() Here is essentially the simplest nontrivial example I know of:Ĭonsider the intersection of the unit circle $\(A)$ but $t\neq 0$ is not the zero function. ![]() ![]() In CPn for any n 1, the line bundle OCPn(1) is positive. This is because the Poincare dual of any single point is the volume form, which is certainly positive. We will discuss a functorial approach to algebraic geometry, leading to the ubiquitous theory of algebraic stacks. if L O( iaipi), where pi are points on C and iai > 0. The very short answer is that the geometry in algebraic geometry comes from considering only polynomial functions as the meaningful functions. If C is a curve, then a line bundle L on C is positive iff L has positive degree, i.e. ) to scheme theory, an extensive bibliography, and also information about the 'art status' of algebraic geometry. This is a big complicated question and many different kinds of answers could be given at many different levels of sophistication. In my humble opinion, the Vakil notes (also known as FOAG) are very complete with regards to scheme theory they include all prerequisites (category theory, commutative algebra, topology, etcetera omissis e.o. (I have a guess, which is that varieties tend to be a lot less tame than manifolds, so you have to jump through more technical hoops to tack on extra stuff to them, but that's pure speculation.) In that case, it would be interesting to try to get a sense of why it's more complicated. Hartshorne has notes on projective geometry which are available online and which I found quite useful. If it turns out the answer is "it's hard to explain, and you just need to read an algebraic geometry text," then that's fine. I found projective geometry confusing when I began learning algebraic geometry. I have to guess that this is a little more complicated than just taking a manifold and adding a metric, otherwise I would expect to be able to find this explained in a relatively straightforward way somewhere. I am assuming that the spaces in this case are algebraic varieties, but what is the extra structure that gets added? What sorts of questions can we answer with this extra structure that we couldn't answer without it? The Nullstellensatz tells you that varieties have points. In other words, the Nullstellensatz is what puts algebra in algebraic geometry. Congratulations on completing this great service to the algebraic geometry community user111072, I guess that your reaction is a result of ignorance about SGA 4 1/2 and its importance in modern algebraic geometry, which is proved further by your mistakenly attributing it to Grothendieck. a text book on algebraic stacks and the algebraic geometry that is needed to. Writing them as C x, y and C ( x, y) with x, y satisfying the algebraic relation E helps seeing what it means. Not everyone likes it, but I do, and routinely recommend it to both undergrads and beginning grad. One place to start, if you are an undergrad, is Miles Reids book Undergraduate Algebraic Geometry. geom., both on this site and on MO, for grad students but also for undergrads. Its fraction field is the field of rational functions E C. Googling will lead you to various roadmaps for learning alg. It is also the ring of polynomial functions E C. Knowing very little about algebraic geometry, I am wondering what the "geometry" part is. Id say that any application of algebraic geometry is an application of the Nullstellensatz as this theorem allows one to use algebraical techniques to study geometry. Contribute to stacks/stacks-project development by creating an account on. X, Y / ( Y X 3 a X b) is the coordinate ring of the elliptic curve E: 3 x b. But I'm sure there is interesting stuff you can ask. In symplectic geometry, we take an even-dimensional manifold and add a symplectic form, and then we can ask about. For example, we can take a smooth manifold and add a Riemannian metric and a connection, and then we can ask about distances between points, curvature, geodesics, etc. We can use it to probe either the structure itself or the underlying space it lives on. The extra structure takes some local data about the space as its input and outputs answers to local or global questions about the space structure. Coming from a physics background, my understanding of geometry (in a very generic sense) is that it involves taking a space and adding some extra structure to it.
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