By G. Ellingsrud, C. Peskine, G. Sacchiero, S. A. Stromme
Algebraic geometers have renewed their curiosity within the interaction among algebraic vector bundles and projective embeddings. New tools were built for questions resembling: what's the geometric content material of syzygies and of bundles derived from them? How can they be used for giving reliable compactifications of ordinary households? Which differential ideas are wanted for the learn of households of projective kinds? those questions are addressed during this cohesive quantity, the place effects, paintings in development, conjectures, and smooth money owed of classical rules are offered.
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Additional info for Complex Projective Geometry: Selected Papers
To is the union of the 5-secants to S. c) Any smooth nonminimal Resurface S of degree 9 with sectional genus 8 is linked (4,4) to a smooth rational surface T of degree 7 and sectional genus 4. S has no 5-secants. d) Any smooth surface of general type S of degree 9 with sectional genus 8 is linked (4,4) to a reducible surface T = To U Po, where To is linked (2,4) to two planes which meet in a point p, and Po is a plane which meets To in two lines through p. Po is the union of the 5-secants to S.
Using the above one can show that any if 3-surface of degree 9 in P 4 can be constructed this way. 28) Remark. The dimension of the family of the surfaces constructed is 48, which equals the Euler characteristic of the normal bundle to 5 in P 4 . 29) Any smooth surface S of general type with TT = 8 is linked (4,4) to a surface T of degree 7 and with ft(T) = 4. 3c). Secondly, we show that it gives rise to a smooth surface of general type of degree 9. On 5 the canonical curves K move in a pencil residual to a plane quartic curve C = H — K.
11) implies that To is smooth outside P . Let JV, = P,; D P i = 1,2,3. Since U2 does not contain P , the intersection V2 C\ P = (iVi UiV2 U N3). Now t/i is singular along P , so the union To U (Pi U P2 U P3) must be singular along the three lines Ni. Thus To meets P along Ni U N2 U iV3 and is singular at the points of intersection Ni H Nj. 19) Lemma. To is an elliptic scroll with three improper double points, and meets each of the planes Pi, P2 and P3 along a quartic curve. Proof. 7) implies that the one-dimensional part of To fl P t , i = 1,2,3 is a quartic curve.
Complex Projective Geometry: Selected Papers by G. Ellingsrud, C. Peskine, G. Sacchiero, S. A. Stromme