1-2-3 theorem for curves on algebraic surfaces

Abstract

Theorem I. Let D be a 1-connected curve with K D nef and p a D Z 2. Let L, M be line bundles on D such that L À 2K D and M À 2K D are both nef. Then the multiplication map H 0 D; L n H 0 D; M 3 H 0 D; L M is surjective except in the following cases:(1) p a D 2 and L M 1 2K D .(2) D contains a curve E with p a E 1, ED À E 1 and L M 1 2K D on E.Here the symbol 1 means the numerical equivalence.Partially supported by Japan Association for Mathematical Sciences.Brought to you by | University of Arizona Authenticated Download Date | 5/27/15 7:17 PM is a decomposition D D 1 D 2 such that D 1 1 0, D 2 1 0. Let k be an integer. A curve D is said to be (numerically) k-connected, if D 1 D 2 Z k holds for any e¨ective decomposition D D 1 D 2 .Lemma 1.1.1. Let D be a k-connected curve and D D 1 D 2 an e¨ective decomposition such that D 1 D 2 k. Then D 1 and D 2 are k 1=2-connected, where x denotes the integer part of x. If D 1 is minimal with respect to the condition D 1 D À D 1 k, then D 1 is k 3=2-connected.Lemma 1.1.2. Let D be a 1-connected curve. Let D i i 1; 2 be a curve contained in D with D i D À D i 1.(1) If D 1 and D 2 have no common components, then either D 1 X D 2 j, or D 1 D 2 1 and D D 1 D 2 .(2) Assume that D 1 and D 2 have a common component. If D 1 is minimal with respect to the condition D 1 D À D 1 1, then D 1 " D 2 .Proof.(1) We have D 1 D 2 " D and D 1 D 2 Z 0, since D 1 and D 2 have no common components. We have Since D is 1-connected, D 1 D 2 D À D 1 À D 2 Z 0 and the equality sign holds here if and only if D D 1 D 2 . Hence either D 1 D 2 0, or D 1 D 2 1, D D 1 D 2 .(2) Let A be the greatest common divisor of D 1 , D 2 , and put D 1 A B, D 2 A C. By the minimality of D 1 , we can assume that C 1 0. We assume that B 1 0 and show that this leads us to a contradiction. Note that A B C " D and A B CD À A À B À C D 1 D À D 1 D 2 D À D 2 À AD À A À 2BC 2 À AD À A À 2BC:We have BC Z 0, since B and C have no common components. Since D 1 is minimal with respect to the condition that D 1 D À D 1 1, we have AD À A Z 2. Since A B CD À A À B À C Z 0, we conclude that AD À A 2, BC 0 and D A B C. On the other hand, since D 1 is 2-connected and D 2 is 1-connected by Lemma 1.1.1, we must have AB Z 2, AC Z 1 which implies AD À A AB C Z 3, a contradiction. Hence B 0 and we have D 1 " D 2 . r A line bundle L on D is called nef if L is of non-negative degree on any component of D. Lemma 1.1.3 ([3]). Let L be a line bundle on D and s A H 0 D; L a non-zero element. Then, either (1) s does not vanish identically on any component of D and L is nef on D, or (2) D is reducible and if Z 0 D is the biggest curve on which s vanishes identically, then Konno, 1-2-3 theorem Brought to you by | University of Arizona Authenticated Download Date | 5/27/15 7:17 PM

Relations in the Canonical Algebras on Surfaces

“…In this appendix, we state some results about canonical algebras of curves on a smooth surface in order to supplement [13].…”

“…2 and K D is nef. In [13], we studied the canonical ring R(D; K D ) L m!0 H 0 (D; mK D ) and showed that it is generated in degrees 3.…”

“…At the end of § 4, we state a result for irregular surfaces similar to Theorem B, (1). In Appendix ( §6), we give a few remarks on the relative canonical algebras for fibrations and singularities, in order to supplement the results in [15] and [13] about Reid's 1-2-3 conjecture.…”

“…Note that, if a curve G i as in (2) exists, then the intersection form is negative semi-definite on Supp (G i ) and G i is nothing more than the numerical cycle on it in the terminology in [20]. The proof of Theorem B is based on the hyperplane section principle as in [19] and a result for chain connected curves similar to those in [15] and [13].…”

On the Fixed Loci of the Canonical Systems over Normal Surface Singularities

“…In Sect. 4, we restrict ourselves to (weakly) elliptic singularities [11] in order to clarify, to some extent, how our method relates to Yau's elliptic sequence [13]. When the fixed part corresponds to a rational double point of type A and the biggest loupe contracts to an elliptic singularity, Theorem 4.1 shows that the associated sequence of loupes is nothing more than the elliptic sequence.…”