## Publication: The Severi bound on sections of rank two semistable bundles on a Riemann surface.

Loading...

##### Files

##### Full text at PDC

##### Publication Date

2001-11

##### Authors

##### Advisors (or tutors)

##### Editors

##### Journal Title

##### Journal ISSN

##### Volume Title

##### Publisher

Princeton University

##### Abstract

Let C be a compact Riemann surface of genus g≥1 . For a line bundle L on C of degree d with 0≤d≤2g−2 , the classical theorem of Clifford states that the dimension h 0 (L) of the space of global sections of L satisfies h 0 (L)≤d 2 +1 . It is also quite easy to list the cases which give equality (apart from the trivial bundle and the canonical bundle, these exist only on hyperelliptic curves). This theorem can be extended to a semistable bundle E on C of rank n and degree d with 0≤d≤n(2g−2) , when it states that h 0 (E)≤d 2 +n . Now let E be a semistable bundle of rank 2 . In this case the authors obtain a more refined result depending on the Segre invariant s(E) , which is defined as the minimum value of degE/L−degL for L a line subbundle of E . (In particular, the semistability condition becomes s(E)≥0 .) The main theorem is that, if degE=d and s≤d≤4g−4−s , then h 0 (E)≤d−s 2 +δ , where δ=1 or δ=2 . (Note that d−s is always even.) In a previous paper, E. Arrondo and the second author conjectured this inequality with δ=1 (with a small number of exceptions similar to those for rank 1 ). However, this is not correct and δ=2 is sometimes necessary; the authors call the bundles which attain this bound Severi bundles. The number δ is obtained as follows. We define polynomials K r (n,N) (the Krawtchouk polynomials) by means of the generating function ∑ r K r (n,N)z r =(1−z) n (1+z) N−n . Let r=d−s 2 +1 ; then we can take δ=1 if K r (g,2g−s)≠0 and δ=2 otherwise. The connection with Krawtchouk polynomials is intriguing, since these are used in coding theory and recent work of T. Johnsen has established links between coding theory and semistable bundles of rank 2 on Riemann surfaces; these links make essential use of the Segre invariant s . The key to the proof of the inequality is the following lemma. Let E be a bundle of rank 2 and degree d on a curve of genus g having at least r+1≤g independent sections. If
K r (g,2g+2r−2−d)≠0, then there is a nonzero section of E vanishing at r points. This is proved using a computation of Chern classes, formal properties of K r (n,N) and the Porteous formula. The authors give examples to show that, for any allowable values of g,s,d , the bound δ=1 can be attained (in fact on a hyperelliptic curve). A good deal is known about the vanishing of K r (n,N) , thus providing a restricted list of candidates for the invariants of Severi bundles, but these values do not all arise from actual bundles. The Severi bundles with d−s=0 are all known, and the authors compute those with d−s=2 and d−s=4 . In particular the bundles obtained by I. Grzegorczyk [Ulam Quart. 3 (1996), no. 2, 41 ff., approx. 3 pp. (electronic) ] are Severi bundles with g=5 , s=4 , d=8 . The authors also obtain sharp bounds for the numbers of independent sections of non-semistable bundles.

##### Description

##### UCM subjects

##### Unesco subjects

##### Keywords

##### Citation

E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of Algebraic Curves, Grundlehren Math. Wiss. 267, Springer-Verlag, New York (1985).
E. Arrondo and I. Sols, Bounding sections of bundles on curves, London Math. Soc. Lecture Note Series 179 (Trieste, 1989/Bergen, 1989), 24–31, Cambridge Univ. Press, Cambridge, 1992.
D. Butler, Families of maximal subbundles for rank two bundles on a curve, Math. Ann. 307 (1997), 29–39.
D. Eisenbud and J. Harris, Divisors on general curves and cuspidal rational curves, Invent. Math. 74 (1983), 371–418.
D. Eisenbud, J. Harris, A simpler proof of the Gieseker-Petri theorem on special divisors, Invent. Math. 74 (1983), 269–280.
W. Fulton and R. Lazarsfeld, On the connectedness of degeneracy loci and special divisors, Acta Math. 146 (1981), 271–283.
P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons, Inc., New York (1978).
P. Griffiths, J. Harris, On the variety of special linear systems on a general algebraic curve, Duke Math. J. 47 (1980), 233–272.
I. Grzegorczyk, All bundles in W 3 2,8 have infinitely many maximal subbundles, Ulam Quart. 3 (1996), 3pp.
G. Hanrot, Résolution effective d'équations diophantiennes: algorithmes et applications, Thèse, Université Bordeaux 1 (1997).
R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer-Verlag, New York (1977).
T. Johnsen, Rank two bundles on algebraic curves and decoding of Goppa codes, preprint. cf.
I. Krasikov and S. Litsyn, On integral zeros of Krawtchouk polynomials, J. Combin. Theory 74 (1996), 71–99.
H. Lange and M. S. Narasimhan, Maximal subbundles of rank two vector bundles on curves, Math. Ann. 266 (1983), 55–72.
M. Mignotte and A. Pethő, On the system of diophantine equations x 2 −6y 2 =−5 and x 2 =2z 2 −1 , Math. Scand. 76 (1995), 50–60.
M. Nagata, On self-intersection number of a section on a ruled surface, Nagoya Math. J. 37 (1970), 191–196.
F. Severi, Sulla clasificazione delle rigate algebrice, Rend. Mat. 2 (1941), 1–32.
R. J. Stroeker and B. M. M. de Weger, On a quartic Diophantine equation, Proc. Edinburgh Math. Soc. 39 (1996), 97–114.
R. J. Stroeker, B. M. M. de Weger, On integral zeroes of binary Krawtchouk polynomials, Nieuw Arch. Wisk. 17 (1999), 175–186.
R. J. Stroeker, B. M. M. de Weger, Solving elliptic diophantine equations: the general cubic case, Acta Arith. 87 (1999), 339–365.
X. J. Tan, Extensions of Clifford's theorem (Chinese), Acta Math. Sinica 31 (1988), 710–720.
X. J. Tan, A counterexample of Arrondo-Sols' conjecture, Chinese Sci. Bull. 37 (1992), 1145–1149.
M. Teixidor I Bigas, Brill-Noether theory for stable vector bundles, Duke Math. J. 62 (1991), 385–400.