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Algebraic geometry examples7/2/2023 ![]() ![]() To understand Siu's work you will also need to learn the analytic approach to algebraic geometry which is introduced in Griffiths and Harris. Rather, you will need to learn geometry! And by geometry, I don't mean the abstract foundations of sheaves and schemes (although these may play a role), I mean specific geometric constructions (blowing up, deformation theory, linear systems, harmonic representatives of cohomology classes - i.e. Indeed, they lie squarely on the same axis of research that the Italians, and Zariski, were interested in, namely, the detailed understanding of the birational geometry of varieties.įurthermore, to understand these results, I don't think that you will particularly need to learn the contents of Eisenbud's book (although by all means do learn them if you enjoy it) Both these results would be of just as much interest to the Italians, or to Zariski, as they are to us today. ![]() However, the questions being studied are (by and large) the same.Īs I commented in another post, two of the most important recent results in algebraic geometry are the deformation invariance of plurigenera for varieties of general type, proved by Siu, and the finite generation of the canoncial ring for varieties of general type, proved by Birkar, Cascini, Hacon, and McKernan, and independently by Siu. The only distinction to me seems to be chronological: modern work was done recently, while classical work was done some time ago. I agree with Donu Arapura's complaint about the artificial distinction between modern and classical algebraic geometry. Is this right? If not, succinctly my question is: how great an influence does classical algebraic geometry have on modern algebraic geometry today? While I suspect that, as with other branches of mathematics, "abstraction was invented to analyze the concrete", with all the emphasis currently given to the understanding of abstract tools, for someone who is not very familiar with the subject (such as myself), it seems that algebraic geometry is a "mixture" of general topology and abstract algebra. Would you need to be familiar with something like the contents of Eisenbud's Commutative Algebra: With a View Toward Algebraic Geometry, or is less needed in reality? (I am familiar with more commutative algebra than that which is covered in Atiyah and MacDonald's *Introduction to Commutative Algebra", but less than that which is covered in Eisenbud's textbook.)Īlso, is modern algebraic geometry concerned with abstractions such as schemes, sheaves, topological spaces, commutative and noncommutative rings etc., or is it just classical algebraic geometry in an abstract form? Perhaps more specifically, to do research in modern algebraic geometry, do you need to be familiar with classical algebraic geometry, or is it possible to think of algebraic geometry as an "abstract language" and do research based just on this perception? one needs to know to do research in (or to learn) modern algebraic geometry. This holds with respect to the singular homology as well as for the Borel-Moore homology.Can anyone offer advice on roughly how much commutative algebra, homological algebra etc. We complement the results in the Turing model by proving, for all k in N, the FPSPACE-hardness of the problem of computing the k-th Betti number of the zet of real zeros of a given integer polynomial. We also define new counting complexity classes in the classical Turing model via taking Boolean parts of the classes above, and show that the problems to compute the Euler characteristic and the geometric degree of (semi)algebraic sets given by integer polynomials are complete in these classes. We prove that the problem of computing the (modified) Euler characteristic of semialgebraic sets is FP_R^-complete. We investigate to what extent the new counting classes capture the complexity of computing basic topological invariants of semialgebraic sets (over R) and algebraic sets (over C). The problems of counting the number of solutions of systems of polynomial inequalities over R, or of systems of polynomial equalities over C, respectively, turn out to be natural complete problems in these classes. ![]() We define counting classes #P_R and #P_C in the Blum-Shub-Smale setting of computations over the real or complex numbers, respectively. ![]()
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