One drawback is that they are very long and they are online notes so there are many typos.
But most of them are grammatical and easy to spot. Algebraic Geometry: A First Course  by Joe Harris is a very good book that sits in that region between undergraduate treatments and the prerequisites of Hartshorne. In particular, one does not need to know much commutative algebra to get a lot out of Harris's book. Harris himself recommends reading Hartshorne after his book for the theory of schemes. I think it is a great introductory textbook to modern algebraic geometry scheme theory.
I found that Mumford is quite good at motivating new concepts; in particular I really enjoy his development of nonsingularity and the sheaf of differentials. I think another great aspect about this book is that it emphasizes how to define things intrinsically i.
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A classic example of the above:. Choose an affine neighborhood so that p corresponds to the origin. Then this affine neighborhood is spec k[x1, Let I' be all the linear terms of I i. Then the tangent space at p is spec k[x1, Some drawbacks. This book doesn't cover nearly as much as Hartshorne's book. It doesn't have that many exercises. Nevertheless I think its a great compliment to reading Hartshorne. The Invitation to algebraic geometry  by Smith et al. I'm really enjoying Andreas Gathmann's lecture notes . They are pretty elementary and surprisingly complete for lecture notes.
I possibly cannot say what is "the best" book in this topic, but I've recently started studying it and found Hartshorne's book extremely difficult, so I went to study Mumford's red book of varieties.
But other than these books that have been introduced I found the followings also helpful:. A Royal Road to Algebraic Geometry  by Audun Holme is a newly published book which tries to make Algebraic Geometry as easy as possible for studetns. Also, the book by Griffits and Harris called Principles of Algebraic Geometry  in spite of being rather old, and working mostly with only complex field, gives a good intuition on this very abstract topic. Again this book covers much less then Hartshorne and only discusses curves over the complex numbers and their Jacobians.
But it gives a lot more details and examples of concepts which I found particularly difficult when I first started learning algebraic geometry sheafs, divisors, cohomology. It also has a bunch of exercises which I think are often not as challenging as the the exercises in Hartshorne. It also covers a lot more of the 'classical' theory of curves than Hartshrone does; e.
Weierstrass points. Best Algebraic Geometry text book? Lifted from Mathoverflow  : I think almost everyone agrees that Hartshorne's Algebraic Geometry is still the best. Hope that is ok, if not, I shall remove it. I was just trying to be complete in the sense that the best book on algebraic geometry besides Hartshorne is not only one, but depends on the level or subject within Algebraic Geometry you are referring to. Instead of posting 13 answers reflecting these distinctions, I just posted all inside one answer.
I hope it is ok? Then there is a bijection between prime ideals of and homogenous prime ideals of. Define the ring.
Algebraic Geometry: Part I Schemes. with Examples and Exercises
Then 1 there is a bijection between homogenous primes of and and 2 this bijection extends to an isomorphism of the corresponding projective schemes. And the second suggestion is already done; it is 7. D in the current version. I presume the hint for exercise 6. And if you solve the problem, please let me know. The part I fear will be too hard is proving that it is not a unique factorization domain. C Helpful to remind about 6. A global assumption could profitably be stated earlier: k denotes a field, except where it clearly stands for a nonnegative?
Could also have default notation k bar for alg. Would improve make tighter MANY future phrases, at your discretion. Perhaps list most notational assumptions in 1. Notion of A-scheme extends it seems without conflict? Say X, as an A-scheme, is loc. D First ref. E is OK, second may have been meant to point a different Ex. E Seems OK, intuitively, but the continuous map like that of 4.
Math 643 Algebraic Geometry I, Fall 2017
G should go the other way. Analytic is already OK over R; add holomorphic and say what more that implies. Only finish later, with c. A bit odd to interrupt proof with Ex. G May mislead to refer to 5. Shouldn't "q. End could also be clearer. D At end: define clearly what I is; cf. H Say f is in A even before a. Assume root is in K B but not in A. Prudent to define the F bar in a. Probably need ref. M OK, recently improved. The main idea doesn't even need this, works for ANY field extension. May want to give a better example — cf.
Maybe reorder: leave no hint that Hint is about version of converse. Say more about Z[sqrt -5 ]: what affine opens work? Maybe give ref. B-H Imp. Of course, still use algebra selectively. But see below. Maybe remove first "geometric axioms" to avoid repeating. Supp uses germs, D f doesn't. Needs work unlike 6. Does NOT seem to be a good illustration of geometric ideas. Bad hint; relies on algebra deeper than needed: to exclude points in Supp f , not in D f bar, must know no minimal primes can contain both f and Ann f. Equivalent: all elements of minimal primes are zerodivisors [AM Ch3 Ex9].
Algebraic Geometry - Part I: Schemes. With Examples and Exercises | Ulrich Görtz | Springer
I did not see how doing i first would help. Another way is below. E To get the locus of nonreduced points closed, not just a union of closures of associated points, do need something Noetherian-like. I did not like the organization of your [E]-based approach, so rethought all, in the following way I believe is shorter and easier to follow.
Is it convincing? Algebra gets initial results. What nontrivial arguments use little algebra? In a set U of points of a scheme, with U open or closed or even constructible, every point of U lies above a U-minimal one, by Zorn. Such points determine the irreducible components of U.
Now work in an affine open Spec A. What makes a section m tilde for M vanish at a point p? Use the ideal Ann m of such elements f, or its radical. The support of m tilde is clearly V Ann m.