Introduction To Gis Textbook

[Francoise Witsell]

Anotheroption would be Welsh's Matroid Theory. This is an older book (it predates even Oxley's first edition), but is nicely written with a more geometric flavour. Also, I just found out that Welsh's book recently became available as a Dover paperback, so it's very cheap if you want to buy it.

My usual recommendations are Theory of Matroids and Matroid Applications. These are multi-author volumes edited by N. White, and he has done a great job choosing authors and homogenizing notation. Although I believe the second volume was intended to be more applied, I didn't really feel there was a difference between the two in this regard.

It is an undergraduate textbook on matroids! (Maybe the first undergraduate one completely devoted to matroids?) And from what I could see from the first few chapters, it is highly pedagogical and very well written.

1) Aigner's classical Combinatorial Theory textbook which has two lengthy and well written chapters on matroids, totaling about 140 pp. Some material is a bit dated (historical notes, refs, etc.) but it's great as an introduction, and recent surveys will fill you in on modern developments.

2) Vol. B of Schrijver's Combinatorial Optimization monograph. It's not really a textbook and it covers only certain topics in matroid theory, but it might be better suited for applications you need, and is terrifically written.

I agree that Tony Huynh's suggestions of Oxley and Welsh are good ones. Depending on what you mean by "suited to engineers," another option might be Lawler's Combinatorial Optimization: Networks and Matroids. Lawler's book focuses on the algorithmic aspects of matroids. Lawler's book was reprinted by Dover but it now seems to be out of print again. :-( However, it seems that you can still find inexpensive copies for sale on the web easily enough.

My first recommendation would be Oxley's Matroid Theory (Oxley J. G., Matroid Theory, Oxford University Press, Oxford, 1992). It is a well organized textbook.If you are looking for application, Recski A., Matroid Theory and its Applications, Springer Verlag, Berlin (1989) would be best option.

I think Algebraic Geometry is too broad a subject to choose only one book. Maybe if one is a beginner then a clear introductory book is enough or if algebraic geometry is not ones major field of study then a self-contained reference dealing with the important topics thoroughly is enough. But Algebraic Geometry nowadays has grown into such a deep and ample field of study that a graduate student has to focus heavily on one or two topics whereas at the same time must be able to use the fundamental results of other close subfields. Therefore I find the attempt to reduce his/her study to just one book (besides Hartshorne's) too hard and unpractical. That is why I have collected what in my humble opinion are the best books for each stage and topic of study, my personal choices for the best books are then:

CLASSICAL:Beltrametti-Carletti-Gallarati-Monti. "Lectures on Curves, Surfaces and Projective Varieties" which starts from the very beginning with a classical geometric style. Very complete (proves Riemann-Roch for curves in an easy language) and concrete in classic constructions needed to understand the reasons about why things are done the way they are in advanced purely algebraic books. There are very few books like this and they should be a must to start learning the subject. (Check out Dolgachev's review.)

HALF-WAY/UNDERGRADUATE:Shafarevich - "Basic Algebraic Geometry" vol. 1 and 2. They may be the most complete on foundations for varieties up to introducing schemes and complex geometry, so they are very useful before more abstract studies. But the problems are hard for many beginners. They do not prove Riemann-Roch (which is done classically without cohomology in the previous recommendation) so a modern more orthodox course would be Perrin's "Algebraic Geometry, An Introduction", which in fact introduce cohomology and prove RR.

ADVANCED UNDERGRADUATE:Holme - "A Royal Road to Algebraic Geometry". This new title is wonderful: it starts by introducing algebraic affine and projective curves and varieties and builds the theory up in the first half of the book as the perfect introduction to Hartshorne's chapter I. The second half then jumps into a categorical introduction to schemes, bits of cohomology and even glimpses of intersection theory.

FREE ONLINE NOTES:Gathmann - "Algebraic Geometry" (All versions are found here. The latest is of 2019.) Just amazing notes; short but very complete, dealing even with schemes and cohomology and proving Riemann-Roch and even hinting Hirzebruch-R-R. It is the best free course in my opinion, to get enough algebraic geometry background to understand the other more advanced and abstract titles. For an abstract algebraic approach, the nice, long notes by Ravi Vakil is found here. (A link to all versions; the latest is of 2017.)

GRADUATE FOR ALGEBRISTS AND NUMBER THEORISTS:Liu Qing - "Algebraic Geometry and Arithmetic Curves". It is a very complete book even introducing some needed commutative algebra and preparing the reader to learn arithmetic geometry like Mordell's conjecture, Faltings' or even Fermat-Wiles Theorem.

GRADUATE FOR GEOMETERS:Griffiths; Harris - "Principles of Algebraic Geometry". By far the best for a complex-geometry-oriented mind. Also useful coming from studies on several complex variables or differential geometry. It develops a lot of algebraic geometry without so much advanced commutative and homological algebra as the modern books tend to emphasize.

BEST ON SCHEMES:Grtz; Wedhorn - Algebraic Geometry I, Schemes with Examples and Exercises. Tons of stuff on schemes; more complete than Mumford's Red Book (For an online free alternative check Mumfords' Algebraic Geometry II unpublished notes on schemes.). It does a great job complementing Hartshorne's treatment of schemes, above all because of the more solvable exercises.

UNDERGRADUATE ON ALGEBRAIC CURVES:Fulton - "Algebraic Curves, an Introduction to Algebraic Geometry" which can be found here. It is a classic and although the flavor is clearly of typed concise notes, it is by far the shortest but thorough book on curves, which serves as a very nice introduction to the whole subject. It does everything that is needed to prove Riemann-Roch for curves and introduces many concepts useful to motivate more advanced courses.

GRADUATE ON ALGEBRAIC CURVES:Arbarello; Cornalba; Griffiths; Harris - "Geometry of Algebraic Curves" vol 1 and 2. This one is focused on the reader, therefore many results are stated to be worked out. So some people find it the best way to really master the subject. Besides, the vol. 2 has finally appeared making the two huge volumes a complete reference on the subject.

INTRODUCTORY ON ALGEBRAIC SURFACES:Beauville - "Complex Algebraic Surfaces". I have not found a quicker and simpler way to learn and clasify algebraic surfaces. The background needed is minimum compared to other titles.

ADVANCED ON ALGEBRAIC SURFACES:Badescu - "Algebraic Surfaces". Excellent complete and advanced reference for surfaces. Very well done and indispensable for those needing a companion, but above all an expansion, to Hartshorne's chapter.

ON HODGE THEORY AND TOPOLOGY:Voisin - Hodge Theory and Complex Algebraic Geometry vols. I and II. The first volume can serve almost as an introduction to complex geometry and the second to its topology. They are becoming more and more the standard reference on these topics, fitting nicely between abstract algebraic geometry and complex differential geometry.

INTRODUCTORY ON MODULI AND INVARIANTS: Mukai - An Introduction to Invariants and Moduli. Excellent but extremely expensive hardcover book. When a cheaper paperback edition is released by Cambridge Press any serious student of algebraic geometry should own a copy since, again, it is one of those titles that help motivate and give conceptual insights needed to make any sense of abstract monographs like the next ones.

ON MODULI SPACES AND DEFORMATIONS:Hartshorne - "Deformation Theory". Just the perfect complement to Hartshorne's main book, since it did not deal with these matters, and other books approach the subject from a different point of view (e.g. geared to complex geometry or to physicists) than what a student of AG from Hartshorne's book may like to learn the subject.

ON GEOMETRIC INVARIANT THEORY:Mumford; Fogarty; Kirwan - "Geometric Invariant Theory". Simply put, it is still the best and most complete. Besides, Mumford himself developed the subject. Alternatives are more introductory lectures by Dolgachev.

ON INTERSECTION THEORY:Fulton - "Intersection Theory". It is the standard reference and is also cheap compared to others. It deals with all the material needed on intersections for a serious student going beyond Hartshorne's appendix; it is a good reference for the use of the language of characteristic classes in algebraic geometry, proving Hirzebruch-Riemann-Roch and Grothendieck-Riemann-Roch among many interesting results.

ON SINGULARITIES:Kollr - Lectures on Resolution of Singularities. Great exposition, useful contents and examples on topics one has to deal with sooner or later. As a fundamental complement check Hauser's wonderful paper on the Hironaka theorem.

ON POSITIVITY:Lazarsfeld - Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear Series and Positivity in Algebraic Geometry II: Positivity for Vector Bundles and Multiplier Ideals. Amazingly well written and unique on the topic, summarizing and bringing together lots of information, results, and many many examples.

INTRODUCTORY ON HIGHER-DIMENSIONAL VARIETIES:Debarre - "Higher Dimensional Algebraic Geometry". The main alternative to this title is the new book by Hacon/Kovcs' "Classifiaction of Higher-dimensional Algebraic Varieties" which includes recent results on the classification problem and is intended as a graduate topics course.

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