Derivatives of Wronskians with applications

to families of special Weierstrass points

Authors:
Letterio Gatto and Fabrizio Ponza

Journal:
Trans. Amer. Math. Soc. **351** (1999), 2233-2255

MSC (1991):
Primary 14H10, 14H15, 14H55, 14H99

DOI:
https://doi.org/10.1090/S0002-9947-99-02343-0

Published electronically:
February 4, 1999

MathSciNet review:
1615963

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a flat proper family of smooth connected projective curves parametrized by some smooth scheme of finite type over . On every such a family, suitable *derivatives* along the fibers" (in the sense of Lax) of the *relative wronskian*, as defined by Laksov and Thorup, are constructed. They are sections of suitable jets extensions of the -th tensor power of the relative canonical bundle of the family itself.

The geometrical meaning of such sections is discussed: the zero schemes of the -th derivative () of a relative wronskian correspond to families of Weierstrass Points (WP's) having weight at least .

The locus in , the coarse moduli space of smooth projective curves of genus , of curves possessing a WP of weight at least , is denoted by . The fact that has the expected dimension for all was implicitly known in the literature. The main result of this paper hence consists in showing that has the expected dimension for all . As an application we compute the codimension Chow (-)class of for all , the main ingredient being the definition of the -th *derivative of a relative wronskian*, which is the crucial tool which the paper is built on. In the concluding remarks we show how this result may be used to get relations among some codimension Chow (-)classes in (), corresponding to varieties of curves having a point with a suitable prescribed Weierstrass Gap Sequence, relating to previous work of Lax.

**[A]**Enrico Arbarello,*Weierstrass points and moduli of curves*, Compositio Math.**29**(1974), 325–342. MR**0360601****[ACGH]**E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris,*Geometry of algebraic curves. Vol. I*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR**770932****[AK]**Allen Altman and Steven Kleiman,*Introduction to Grothendieck duality theory*, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin-New York, 1970. MR**0274461****[Co1]**Marc Coppens,*The number of Weierstrass points on some special curves. I*, Arch. Math. (Basel)**46**(1986), no. 5, 453–465. MR**847090**, https://doi.org/10.1007/BF01210786**[Co2]**M. Coppens, Private communication, (1997).**[Cu]**Fernando Cukierman,*Families of Weierstrass points*, Duke Math. J.**58**(1989), no. 2, 317–346. MR**1016424**, https://doi.org/10.1215/S0012-7094-89-05815-8**[D1]**Steven Diaz,*Exceptional Weierstrass points and the divisor on moduli space that they define*, Mem. Amer. Math. Soc.**56**(1985), no. 327, iv+69. MR**791679**, https://doi.org/10.1090/memo/0327**[D2]**Steven Diaz,*Tangent spaces in moduli via deformations with applications to Weierstrass points*, Duke Math. J.**51**(1984), no. 4, 905–922. MR**771387**, https://doi.org/10.1215/S0012-7094-84-05140-8**[DM]**P. Deligne and D. Mumford,*The irreducibility of the space of curves of given genus*, Inst. Hautes Études Sci. Publ. Math.**36**(1969), 75–109. MR**0262240****[EH]**David Eisenbud and Joe Harris,*When ramification points meet*, Invent. Math.**87**(1987), no. 3, 485–493. MR**874033**, https://doi.org/10.1007/BF01389239

David Eisenbud and Joe Harris,*Existence, decomposition, and limits of certain Weierstrass points*, Invent. Math.**87**(1987), no. 3, 495–515. MR**874034**, https://doi.org/10.1007/BF01389240**[Fa1]**Carel Faber,*Chow rings of moduli spaces of curves. I. The Chow ring of \overline{ℳ}₃*, Ann. of Math. (2)**132**(1990), no. 2, 331–419. MR**1070600**, https://doi.org/10.2307/1971525

Carel Faber,*Chow rings of moduli spaces of curves. II. Some results on the Chow ring of \overline{ℳ}₄*, Ann. of Math. (2)**132**(1990), no. 3, 421–449. MR**1078265**, https://doi.org/10.2307/1971526**[Fa2]**C. Faber,*A Conjectural Description of the Moduli Space of Curves*, Amsterdam-Utrecht seminar on moduli spaces of curves" (C. Faber, E.Looijenga, eds.), A. & K. Peters Inc., to appear.**[Fo]**O. Forster,*Lectures on Riemann Surfaces*, Springer-Verlag, 1984.**[Fu]**William Fulton,*Intersection theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR**732620****[Ga1]**Letterio Gatto,*Weight sequences versus gap sequences at singular points of Gorenstein curves*, Geom. Dedicata**54**(1995), no. 3, 267–300. MR**1326732**, https://doi.org/10.1007/BF01265343**[Ga2]**Letterio Gatto,*Weierstrass loci and generalizations. I*, Projective geometry with applications, Lecture Notes in Pure and Appl. Math., vol. 166, Dekker, New York, 1994, pp. 137–166. MR**1302947****[Ga3]**L. Gatto,*On the closure in of smooth curves having a special Weierstrass point*, Math. Scand., to appear.**[GP]**L. Gatto, F. Ponza,*Generalized Wronskian Sections and Families of Weierstrass Points with Prescribed Minimal Weight*, Rapporto Interno, Politecnico di Torino, n. 31 (1996), (hard copies available upon request).**[Gu]**R. C. Gunning,*Lectures on Riemann surfaces*, Princeton Mathematical Notes, Princeton University Press, Princeton, N.J., 1966. MR**0207977****[Ha]**Robin Hartshorne,*Algebraic geometry*, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR**0463157****[HM]**Joe Harris and David Mumford,*On the Kodaira dimension of the moduli space of curves*, Invent. Math.**67**(1982), no. 1, 23–88. With an appendix by William Fulton. MR**664324**, https://doi.org/10.1007/BF01393371**[La]**Dan Laksov,*Weierstrass points on curves*, Young tableaux and Schur functors in algebra and geometry (Toruń, 1980), Astérisque, vol. 87, Soc. Math. France, Paris, 1981, pp. 221–247. MR**646822****[L1]**R. F. Lax,*Weierstrass points of the universal curve*, Math. Ann.**216**(1975), 35–42. MR**0384809**, https://doi.org/10.1007/BF02547970**[L2]**Robert F. Lax,*Gap sequences and moduli in genus 4*, Math. Z.**175**(1980), no. 1, 67–75. MR**595632**, https://doi.org/10.1007/BF01161382**[LT1]**Dan Laksov and Anders Thorup,*The Brill-Segre formula for families of curves*, Enumerative algebraic geometry (Copenhagen, 1989) Contemp. Math., vol. 123, Amer. Math. Soc., Providence, RI, 1991, pp. 131–148. MR**1143551**, https://doi.org/10.1090/conm/123/1143551**[LT2]**Dan Laksov and Anders Thorup,*Weierstrass points and gap sequences for families of curves*, Ark. Mat.**32**(1994), no. 2, 393–422. MR**1318539**, https://doi.org/10.1007/BF02559578**[Mu1]**David Mumford,*Stability of projective varieties*, Enseignement Math. (2)**23**(1977), no. 1-2, 39–110. MR**0450272****[Mu2]**David Mumford,*Towards an enumerative geometry of the moduli space of curves*, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 271–328. MR**717614****[OS]**Frans Oort and Joseph Steenbrink,*The local Torelli problem for algebraic curves*, Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, pp. 157–204. MR**605341****[Pa]**Charles Patt,*Variations of Teichmueller and Torelli surfaces*, J. Analyse Math.**11**(1963), 221–247. MR**0160894**, https://doi.org/10.1007/BF02789986**[Pi]***Deformation of Algebraic Varieties with -action*, (Astérisque 2a).**[Po1]**F. Ponza,*Sezioni wronskiane generalizzate e famiglie di punti di Weierstrass*, Doctoral Thesis, Consorzio Universitario Torino-Genova, 1997.**[Po2]**F. Ponza,*Generalized Wronskian Sections and their Zero-Loci*, in preparation (1997).**[R]**H. E. Rauch,*Weierstrass points, branch points, and moduli of Riemann surfaces*, Comm. Pure Appl. Math.**12**(1959), 543–560. MR**0110798**, https://doi.org/10.1002/cpa.3160120310

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
14H10,
14H15,
14H55,
14H99

Retrieve articles in all journals with MSC (1991): 14H10, 14H15, 14H55, 14H99

Additional Information

**Letterio Gatto**

Affiliation:
Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24-10129 Torino, Italy

Email:
lgatto@polito.it

**Fabrizio Ponza**

Affiliation:
Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24-10129 Torino, Italy

Email:
ponza@dm.unito.it

DOI:
https://doi.org/10.1090/S0002-9947-99-02343-0

Keywords:
Relative wronskians,
derivatives of relative wronskians,
families of Weierstrass points,
moduli spaces of curves,
Chow classes in $M_{g}$

Received by editor(s):
February 5, 1997

Published electronically:
February 4, 1999

Additional Notes:
Work partially supported by GNSAGA-CNR, MURST and by Dottorato di Ricerca in Matematica, Consorzio Universitario Torino-Genova

Article copyright:
© Copyright 1999
American Mathematical Society