William Fulton's Algebraic Curves: An Introduction to Algebraic Geometry PDF

By William Fulton

ISBN-10: 0201510103

ISBN-13: 9780201510102

ISBN-10: 0805330828

ISBN-13: 9780805330823

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Let p > 0, and let K be a fibered knot. Then Kp,q has a Seifert surface which is isotopic to a piece of a complex curve Vf ∩ B 4 if and only if • K has a Seifert surface which is isotopic to a piece of a complex curve and • q>0 In particular, the fiber surface of an iterated torus knot is isotopic to a piece of a complex curve if and only if all the cabling coefficients are positive. 5. 10 of [BaaIsh]). They asked whether there exists a quasipositive fiber surface, other than a disk, from which we cannot deplumb a Hopf band (see Figure 1 and the associated discussion in Section 2 for relevant definitions).

Then for each characteristic line bundle L on X such that the formal dimension of the Seiberg-Witten moduli space is − m non-negative even integer 2m, then SW+ X (L) − SWX (L) = −(−1) . Note that when b− 2 ≤ 9, it follows from the above result that there is well defined Seiberg-Witten invariant which will be denoted as SWoX (X). 4. [T] Suppose that X is closed symplectic 4-manifold with b+ 2 (X) > 1 − (X) = 1). If K is a canonical class of X, then SW (±K ) = ±1 (SW (b+ X X X 2 X (−KX ) = ±1). 2.

Indeed, this could probably be proved with the rudiments of convex surface theory. We content ourselves here with the case of S 3 since the results follow easily from our previous 51 HEDDEN work on cables. Moreover, our proof serves to highlight the rich geometric content of the knot Floer homology invariants. 4. Acknowledgment: This note draws on several aspects of a talk I gave at the 2007 G¨okova Geometry and Topology conference. I wish to thank the organizers of the conference for their invitation to speak and attend, and the NSF and IMU for providing funding.

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Algebraic Curves: An Introduction to Algebraic Geometry by William Fulton

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