# Best Algebraic Geometry message publication? (apart from Hartshorne )

Lifted from Mathoverflow :

I assume (virtually ) every person concurs that Hartshorne's Algebraic Geometry is still the most effective. After that what could be the 2nd ideal? It can be a publication, preprint, on-line lecture note, page, and so on

. One pointer per solution please. Additionally, please include a description of why you like guide, or what makes it one-of-a-kind or valuable.

The Invitation to algebraic geometry by Smith et al. is additionally really legible.

I'm actually appreciating Andreas Gathmann's lecture notes. They are rather primary and also remarkably full (for lecture notes).

Reid additionally has an actually wonderful message on algebraic geometry (" Undergraduate algebraic geometry").

Algebraic Geometry: A First Course by Joe Harris is a great publication that beings in that area in between undergraduate therapies and also the requirements of Hartshorne. Specifically, one does not require to recognize much commutative algebra to get a whole lot out of Harris's publication. Harris himself advises reviewing Hartshorne after his publication for the concept of systems.

Before Hartshorne's publication there was Mumford's Red Book of Varieties. I assume it is a wonderful initial book to modern-day algebraic geometry (system concept).

I located that Mumford is fairly efficient encouraging new principles ; specifically I actually appreciate his growth of nonsingularity and also the sheaf of differentials. I assume an additional wonderful facet concerning this publication is that it stresses just how to specify points inherently (i.e. without reference to a shut or open immersion right into affine room) yet additionally clarifies just how to make neighborhood debates (i.e. making use of immersion right into affine room). A timeless instance of the above :

(non inherent tangent room) : Say X is a selection and also p is a factor of X. Choose an affine area to make sure that p represents the beginning. After that this affine area is spec k [x1, ..., xn ]/ I for some perfect. Allow I' be all the straight regards to I (i.e. if I = (x, y ^ 2), after that I' = (x) ). After that the tangent room at p is spec k [x1, ..., xn ]/ I'.

(inherent tangent room) : Let m be the topmost perfect of the neighborhood ring of the framework sheaf at p, after that the tangent room is the twin of the vector room m/m ^ 2.

Taking specification of the symmetrical algebra of the last offers you the previous.

Some downsides. This publication does not cover virtually as high as Hartshorne's publication. It does not have that several workouts. The notation is a little various ; indispensable limited type systems are called pre - selections and also you can remove the' pre' if it's additionally divided. However I assume its a wonderful praise to reviewing Hartshorne.

Another publication I desire I had actually found out about when I was first analysis Hartshorne is Miranda's Complex Algebraic Curves.

Once more this publication covers a lot less after that Hartshorne and also just reviews contours over the intricate numbers (and also their Jacobians). Yet it offers a whole lot even more information and also instances of principles which I located specifically hard when I first began finding out algebraic geometry (sheafs, divisors, cohomology). It additionally has a number of workouts which I assume are usually not as tough as the the workouts in Hartshorne.

It additionally covers a whole lot even more of the 'timeless' concept of contours than Hartshrone does ; as an example Weierstrass factors.

for *Undergraduate * algebraic geometry (dramatically listed below the degree of Hartshorne), Cox, Little and also O'Shea's *Ideals, Varieties, and also Algorithms * is a positive therapy.

My last pointer would certainly be Ravi Vakil's online notes on the foundations of algebraic geometry.

I assume these notes could be made right into a complete on book sooner or later. I have not browsed every one of them yet these notes appear to cover as high as Hartshorne does (otherwise even more). Just hardly ever do Hartshorne and also Vakil specify points in different ways (' projective morphisms' is the only instance that enters your mind).

I've heard it claimed that Hartshorne's publication is a' child 'variation of EGA. I assume Vakil's notes are someplace in between Hartshorne and also EGA (possibly not the midpoint though). At the very least Vakil reviews far more the concept of representable functors, and also Noetherian theory are much less widespread in Vakil's notes. Additionally Vakil's notes are extra full because they additionally include evidence of most of the commutative algebra results that are simply mentioned in Hartshorne.

I assume Vakil invests a whole lot even more time encouraging the product and also usually the notes are a little bit conversational. Additionally there are lots of workouts and also a lot of the them are added with valuable qualifiers like (very easy yet vital workouts, useless workout, laborious yet valuable workout, etc).

One downside is that they are long and also they are on-line notes so there are several typos. Yet a lot of them are grammatic and also very easy to place. [Modify : By currently there are just a couple of typos (* due to the fact that * these are on-line notes) ]

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