By Julian Lowell Coolidge

Students and lecturers will welcome the go back of this unabridged reprint of 1 of the 1st English-language texts to provide complete insurance of algebraic aircraft curves. It deals complex scholars an in depth, thorough advent and heritage to the idea of algebraic aircraft curves and their family members to numerous fields of geometry and analysis.

The textual content treats such issues because the topological houses of curves, the Riemann-Roch theorem, and all features of a wide selection of curves together with genuine, covariant, polar, containing sequence of a given style, elliptic, hyperelliptic, polygonal, reducible, rational, the pencil, two-parameter nets, the Laguerre internet, and nonlinear platforms of curves. it really is nearly solely constrained to the homes of the overall curve instead of an in depth examine of curves of the 3rd or fourth order. The textual content mainly employs algebraic process, with huge parts written in response to the spirit and strategies of the Italian geometers. Geometric equipment are a lot hired, although, specifically these regarding the projective geometry of hyperspace.

Readers will locate this quantity plentiful guidance for the symbolic notation of Aronhold and Clebsch.

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**Example text**

X1 x22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x31 x2 • 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

X1 5 x1 • Dickson’s Lemma can be generalized to monomial modules as follows. 9. (Structure Theorem for Monomial Modules) Let M ⊆ P r be a monomial module. e. there are finitely many terms t1 , . . , ts ∈ Tn and numbers γ1 , . . , γs ∈ {1, . . , r} such that we have M = ht1 eγ1 , . . , ts eγs i. b) There are monomial ideals I1 , . . , Ir ⊆ P such that M is of the form M∼ = ⊕ri=1 Ii ei .

B) An element f ∈ R is the least common multiple of f1 , . . , fm if and only if fi | f for i = 1, . . , m and every element g ∈ R such that fi | g for i = 1, . . , m satisfies f | g . Q Proof. First we prove a). For i = 1, . . QUsing the definition and induction on m, we see that gcd(f1 , . . ,αpm } . Thus it follows immediately that gcd(f1 , . . , fm ) divides fi for i = 1, . . , m. Q Now let g ∈ R be a common divisor of f1 , . . , fm , and let g = c p∈P pβp be the factorization of g .

### A treatise on algebraic plane curves by Julian Lowell Coolidge

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