Algebraic Geometry I: Complex Projective Varieties (Classics by David Mumford

By David Mumford

From the stories: "Although a number of textbooks on smooth algebraic geometry were released meanwhile, Mumford's "Volume I" is, including its predecessor the purple booklet of sorts and schemes, now as earlier than essentially the most first-class and profound primers of contemporary algebraic geometry. either books are only precise classics!" Zentralblatt

Show description

Read Online or Download Algebraic Geometry I: Complex Projective Varieties (Classics in Mathematics) PDF

Similar algebraic geometry books

Hodge theory and complex algebraic geometry 1

This can be a sleek advent to Kaehlerian geometry and Hodge constitution. insurance starts with variables, advanced manifolds, holomorphic vector bundles, sheaves and cohomology concept (with the latter being taken care of in a extra theoretical approach than is common in geometry). The ebook culminates with the Hodge decomposition theorem.

Logarithmic Forms and Diophantine Geometry

There's now a lot interaction among reviews on logarithmic types and deep facets of mathematics algebraic geometry. New gentle has been shed, for example, at the recognized conjectures of Tate and Shafarevich in terms of abelian kinds and the linked celebrated discoveries of Faltings developing the Mordell conjecture.

Geometry of Higher Dimensional Algebraic Varieties (Oberwolfach Seminars)

The topic of this e-book is the type idea and geometry of upper dimensional kinds: lifestyles and geometry of rational curves through attribute p-methods, manifolds with damaging Kodaira measurement, vanishing theorems, idea of extremal rays (Mori theory), and minimum types. The e-book offers a state of the art intro- duction to a tough and never effortlessly available topic which has passed through huge, immense improvement within the final twenty years.

A Survey of Knot Theory

Knot concept is a quickly constructing box of study with many purposes, not just for arithmetic. the current quantity, written through a widely known expert, supplies an entire survey of this thought from its very beginnings to latest most up-to-date examine effects. An critical e-book for everybody thinking about knot thought.

Additional resources for Algebraic Geometry I: Complex Projective Varieties (Classics in Mathematics)

Example text

10). This completes the proof. The next entry can be found in Slater’s compendium [262, equation (35)]. 6 (p. 35). 8). Then ∞ (−q; q 2 )n q (n+1)(n+2)/2 qf (−q, −q 7 ) . 2 and Auxiliary Results 33 Proof. 1), set h = 2, a = c = −q 2 , t = −q, and b = 0. Multiplying both sides of the resulting identity by q/(1 + q), we find that ∞ ∞ (−q; q 2 )n q (n+1)(n+2)/2 (−q)n q(−q; q 2 )∞ = . 8) three times altogether, we find that ∞ ∞ qn 1 q (n−1)/2 = (1 − (−1)n ) (q; q)2n+1 2 n=0 (q; q)n n=0 1 = √ 2 q 1 1 − √ √ ( q; q)∞ (− q; q)∞ 1 (−q 1/2 ; q 2 )∞ (−q 3/2 ; q 2 )∞ (q 2 ; q 2 )∞ = √ 2 q(q; q)∞ −(q 1/2 ; q 2 )∞ (q 3/2 ; q 2 )∞ (q 2 ; q 2 )∞ 1 = √ 2 q(q; q)∞ 1 =√ q(q; q)∞ = = 1 (q; q)∞ ∞ 2 qn −n/2 ∞ − 2 (−1)n q n −n/2 n=−∞ n=−∞ ∞ (2n+1)2 −(2n+1)/2 q n=−∞ ∞ 4n2 +3n q n=−∞ f (q, q 7 ) .

14), we find that ∞ ∞ ∞ (−1)n q (n+1)(n+2)/2 (−1)n q (n+1)(n+2)/2+m(2n+1) = 2n+1 (q)n (1 − q ) (q)n n=0 n=0 m=0 ∞ ∞ q m+1 = m=0 ∞ (−1)n q n(n−1)/2+n(2m+2) (q)n n=0 q m+1 (q 2m+2 )∞ = m=0 ∞ = q(q)∞ qm (q)2m+1 m=0 = qf (q, q 7 ), which completes the proof. 8 can be found in Slater’s paper [262, equation (37)]. 8 (p. 35). 8), then ∞ (−q; q 2 )n q n(n+1)/2 f (−q 3 , −q 5 ) . 15) Proof. 1), we set h = 2, a = −q 2 , b = 0, and c = t = −q. Upon using Euler’s identity, we find that ∞ ∞ (−q; q 2 )n q n(n+1)/2 (−q)n 2 2 = (−q; q) (−q; q ) .

8), we find that S= (iq 2 ; q 2 )∞ (−i1/2 q; q)∞ (−i−1/2 q; q)∞ 2(−q; q 2 )∞ (iq 2 ; q 2 )∞ +(i1/2 q; q)∞ (i−1/2 q; q)∞ = 1 2(q; q)∞ (−q; q 2 )∞ (−i1/2 ; q)∞ (−i−1/2 q; q)∞ (q; q)∞ 1 + i1/2 + = 1 2(q; q)∞ (−q; q 2 )∞ 1 1 + i1/2 + = (i1/2 ; q)∞ (i−1/2 q; q)∞ (q; q)∞ 1 − i1/2 1 1 − i1/2 1 2(1 − i)(q; q)∞ (−q; q 2 )∞ ∞ in/2 q n(n−1)/2 n=−∞ ∞ (−1)n in/2 q n(n−1)/2 n=−∞ ∞ (1 − i1/2 ) in/2 q n(n−1)/2 n=−∞ ∞ +(1 + i1/2 ) (−1)n in/2 q n(n−1)/2 n=−∞ = 1 (1 − i)(q; q)∞ (−q; q 2 )∞ ∞ ∞ in q n(2n−1) − n=−∞ in+1 q n(2n+1) n=−∞ 50 2 The Sears–Thomae Transformation = ∞ 1 (1 − i)(q; q)∞ (−q; q 2 )∞ ∞ in q n(2n−1) − i (−i)n q n(2n−1) n=−∞ , n=−∞ where we replaced n by −n in the latter sum.

Download PDF sample

Rated 4.08 of 5 – based on 13 votes