Introduction to Plane Algebraic Curves by Ernst Kunz

By Ernst Kunz

This paintings treats an advent to commutative ring thought and algebraic airplane curves, requiring of the scholar just a uncomplicated wisdom of algebra, with all the algebraic evidence accumulated into numerous appendices that may be simply mentioned, as needed.

Kunz's confirmed perception of training subject matters in commutative algebra including their purposes to algebraic geometry makes this booklet considerably diverse from others on aircraft algebraic curves. The exposition makes a speciality of the in basic terms algebraic facets of aircraft curve thought, leaving the topological and analytical viewpoints within the history, with basically informal references to those matters and recommendations for extra reading.

Most vital to this text:

* Emphasizes and makes use of the idea of filtered algebras, their graduated jewelry and Rees algebras, to infer simple evidence in regards to the intersection thought of airplane curves

* provides residue idea within the affine aircraft and its functions to intersection theory

* tools of facts for the Riemann–Roch theorem comply with the presentation of curve thought, formulated within the language of filtrations and linked graded rings

* Examples, workouts, figures and proposals for additional learn around out this relatively self-contained textbook

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From a evaluate of the German edition:

"[T]he reader is invited to benefit a few themes from commutative ring thought through customarily learning their illustrations and functions in airplane curve conception. This methodical method is unquestionably very enlightening and effective for either lecturers and students…The entire textual content is a true masterpiece of readability, rigor, comprehension, methodical ability, algebraic and geometric motivation…highly enlightening, motivating and exciting on the related time…One easily can't do higher in writing this kind of textbook."

--Zentralblatt MATH

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Example text

It contains less information than the intersection scheme F1 ∩ F2 , but more than V+ (F1 ) ∩ V+ (F2 ). We have now arrived at the main theorem of this chapter, whose proof is quite easy by the above. 7. For two curves F1 and F2 in P2 (K) with no common component we have deg(F1 ⋆ F2 ) = deg F1 · deg F2 . Two curves F1 , F2 in P2 (K) always intersect in deg F1 · deg F2 points if F1 and F2 have no common component and their intersection points are counted with the appropriate intersection multiplicities.

We call r a rational function on the projective plane, whose domain of definition, Def(r), is P2 (K)\V+ (ψ). We call ψ the pole divisor and φ the zero divisor of r. The difference φ − ψ in the divisor group D of P2 (K) is called the principal divisor belonging to r. The principal divisors of rational functions on P2 (K) form a subgroup H of D. The residue class group Cl(P2 ) = D/H is called the divisor class group of P2 (K). Using the degree of a divisor, it is easy to show that Cl(P2 ) ∼ = (Z, +).

We can assume that P = (0, 0), and then we have to show that K[f ]MP is a discrete valuation ring. Let f = cf1n1 · · · fhnh be a decomposition of f into irreducible polynomials (c ∈ K ∗ , ni > 0). Since P is a regular point of F , we have fi (P ) = 0 for exactly one i ∈ {1, . . 7(a)). Therefore, (f ) · K[X, Y ]MP = (fi ) · K[X, Y ]MP . Because mP (F ) = 1, we have deg Lf = deg Lfi = 1. , the tangent to F at P is the X-axis. Then fi is of the form 6 Regular and Singular Points of Algebraic Curves.

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