Algebraic Geometry — Open Problems: Proceedings of the Conference Held in Ravello, May 31 – June 5, 1982

418 Pages·1983·4.049 MB·English-French

by  E. Ballico, Ph. Ellia (auth.), Ciro Ciliberto, Franco Ghione, Ferruccio Orecchia (eds.)

Lecture Notes ni
Mathematics
Edited by .A Dold dna .B nnamkcE
997
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sgnideecorP fo eht Conference
Held in .,ollevaR yaM 13 -June ,5 1982
Edited by .C Ciliberto, E Ghione, dna E Orecchia
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Berlin Heidelberg New York Tokyo 1983
Editors
Ciro Ciliberto
Istituto di Matematica .R" Caccioppoli", Universita di Napoli
Via Mezzocannone 8, 80100 Napoli, Italy
Franco Ghione
Dipartimento di Matematica, Universita di Roma II
Tot Vergata, 00100 Roma, Italy
Ferruccio Orecchia
Istituto di Matematica "R. Caccioppoli", Universit& di Napoli
Via Mezzocannone 8, 80100 Napoli, Italy
AMS Subject Classifications (1980): 14-06
ISBN 3-54042320-2 Springer-Verlag Berlin Heidelberg New York Tokyo
ISBN 0-38?-12320-2 Springer-Verlag New York Heidelberg Berlin Tokyo
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© by Springer-Verlag Berlin Heidelberg 1983
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Introduction
The Conference on " Open problems in Algebraic Geometry " was
held in Ravello ( Salerno ) during the week : May 31 st- June 5 th ,
1982 . This volume contains most of the lectures and talks given du-
ring the Conference as well as papers grown up from discussions among
participants The only exception is paper n ° 15 , which however fits
very well with the theme of the Conference . We are extremely grateful
to all participants and in particular to all contributors of this vo-
lume . We also thank :
- the " Consiglio Nazionale delle Ricerche " , the University of Na-
pies , the " Banco di Napoli " , for their financial support ;
- the " Ente Provinciale per il Turismo di Salerno " for having allo-
wed the use of the wonderful " Villa Rufolo " , where the Conferen-
ce took place .
We finally thank all those persons who helped either in the organiza-
tion or in the developement of the Conference .
Ciro ciliberto
Franco Ghione
Ferruccio Orecchia
List of Lectures
A. Conte : " Enriques threefolds "
S. Greco : " Remarks on the singularities of algebraic surfaces "
C. Peskine : " Classification of curves in ~3 ,,
W. Fulton : " Nodal curves "
E. Arbarello : " A few things about the variety of irreducible plane
curves of given degree and genus "
D. Mumford : " The geometry of the moduli space of curves "
F. Catanese : " On the moduli space of surfaces of general type "
S. L. Kleiman : " The enumeration of varieties touching given ones "
A. Beauville : " Problems on rational and unirational varieties "
C. De Concini : " On complete symmetric varieties "
E. Sernesi : " On a problem of uniqueness for certain linear series "
J. Harris : " Problems in the projective geometry of curves "
List of Talks
V. Mehta : " Vector bundles on projective varieties and their
restriction to curves "
C. Ceresa : " Remarks on algebraic equivalence "
n
P. Maroscia : " The Hilbert function of a finite set of points in
G. Van der Geer : " The geometry of a Siegel modular threefold "
R. Gupta : "Schubert calculus and geometry of representation theory "
R. Smith : " The branch locus of the Prym map "
M. Beltrametti : " Conic bundles on non rational surfaces "
H. Hulek : " The normal bundle of space curves "
D. Eisenbud : " Rational curves with cusps "
3 11
E. Stagnaro : " Constructing Enriques surfaces from quintics in P
P. Craighiero : " Cubic surfaces in A 3 whose curves are set-
C
theoretic complete intersection "
List of Participants
A. Albano ( University of Torino ) , E. Ambrogio ( Torino ) , E. Arba-
rello ( Roma I ) , D° Arezzo ( Genova ) , M.G. Ascensi ( Brandeis ) ,
F. Baldassarri ( Padova ) , E. Ballico ( Pisa ) , U. Bartocci ( Peru-
gia ) , A. Beauville ( Ec. Polytechnique,Paris ) , G. Beccari ( Tori-
no ) , B. Bellaccini ( Siena ) , M. Beltrametti ( Genova ) , J.F. Bo-
utot ( Strasbourg ) , M. Brundu ( Genova ) , M. Candilera ( Padova ),
F. Catanese ( Pisa ) , M. Cavaliere ( Genova ) , G. Ceresa ( Torino),
L. Chiantini ( Torino ) , S. Chiaruttini ( Padova ) , Ciro Ciliberto
( Napoli ) , A. Collino ( Torino ) , A. Conte ( Torino ) , M. Contes-
sa ( Roma I ) , M. Cornalba ( Pavia ) , P.C. Cralghiero ( Padova ) ,
V. Cristante ( Padova ) , C. Cumino ( Torino ) , C. De Concini ( Roma
II ) , A. Del Centina ( Firenze ) , P. De Vito (Napoli ) ,Ao Di Sante
( Napoli ) , D. Eisenbud ( Brandeis ) , P. Ellia ( Nice ) , D. Epema
( Leiden ) , G. Faltings ( Wuppertal ) , M. Fiorentini ( Ferrara ) ,
M° Formisano ( Napoli ) , P. Francia ( Genova ) , W. Fulton ( Brown),
S. Gabelli ( Roma I ) , R. Gattazzo ( Padova ) , F. Ghione ( Roma II),
A. Gimigliano ( Firenze ) , S° Greco ( Torino ) , Guerra ( Perugia ),
R. Gupta ( M.I.T° ) , J. Harris ( Brown ) , D. Husemoller , K. Hulek
( Erlangen ) , M. Id~ ( Bologna ) , S. Kleiman ( M.I.T. ) , W. Klein-
ert ( East Berlin ) , Kooler ( Brandeis ) , A. Lanteri ( Milano ) ,
E. Li Marzi ( Messina ) , R. Maggioni ( Catania ) , M. Manaresi ( Bo-
logna ) , M°G. Marinari ( Genova ) , P. Maroscia ( Roma I ) , G. Mar-
tens ( Erlangen ) , C. Martinengo ( Genova ) , C. Massaza ( Siena ) ,
L. Mazzi ( Torino ) , V. Mehta ( Bombay ) , I. Morrison , D. Mumford
( Harvard ) , , G° Niesi ( Genova ) , F. Odetti ( Genova ) , P. Oli-
verio ( Pisa ) , A. Oneto ( Genova ) , F. Orecchia ( Napoli ) , M.
Palleschi ( Parma ) , G. Paxia ( Catania ) , C. Pedrini ( Genova ) ,
U. Persson ( Stockolm ) , C. Peskine ( Oslo ) , L. Picco Botta ( To-
rino ) , A. Ragusa ( Catania ) , L. Ramella ( Genova ) , S. Recillas
( Firenze ) , L. Robbiano ( Genova ) , N. Rodin8 ( Firenze ) , M. Rog-
gero ( Genova ) , D° Romagnoli ( Torino ) , M.E. Rossi ( Genova ) , G.
Sacchiero ( Ferrara ) , P. Salmon ( Genova ) , F.O. Sehreyer ( Bran,
dels ) , E. Sernesi ( Roma I ) , M.E. Serpico ( Genova ) , J. Shah (
Northeastern ) , R. Smith ( Georgia ) , E. Stagnaro ( Padova ) , R.
Strano ( Catania ) , E. Strickland ( Roma I ) F. Sullivan ( Pado-
va ) , G. Tamone ( Genova ) , G. Tedeschi ( Torino ) , C. Traverso (
Pisa ) , C. Turrini ( Milano ) , Ughi ( Perugia ) , G. Valla ( Genova),
Go Van de Geer ( Amsterdam ) , , B. Van Geemen ( Utrecht ) , G. Vec-
chio ( Catania ) , L. Verdi ( Firenze ) , A. Verra ( Torino ) , G.E.
Welters ( Barcelona )
TABLE OF CONTENTS
E. BALLICO and Ph. ELLIA:
On degeneration of projective curves ...................... i
A. BEAUVILLE:
Vari~t~s rationelles et unirationelles .................... 16
M. BELTRAMETTI and P. FRANCIA:
Conic bundles on non-rational surfaces .................... 34
F. CATANESE:
Moduli of surfaces of general type ........................ 90
C. CILIBERTO:
On a proof of Torelli's theorem ........................... 113
A. CONTE:
Two examples of algebraic threefolds whose hyperplane
sections are Enriques surfaces ............................ 124
D. EISENBUD and J. HARRIS:
On the Brill - Noether theorem ............................ 131
G. FALTINGS:
Properties of Arakelov's intersection product ............. 138
W. FULTON:
On nodal curves ........................................... 146
W. FULTON, S. KLEIMAN and R. MACPHERSON:
About the enumeration of contacts ......................... 156
F. GHIONE:
Un probleme du type Brill-Noether pour les
fibres vectoriels ........................................ 197
S. GRECO and A. VISTOLI:
On the construction of rational surfaces with
assigned singularities .................................... 210
L. GRUSON and C. PESKINE:
Postulation des courbes gauches ........................... 218
lllV
K. HULEK:
Projective geometry of elliptic curves ....................... 228
R. LAZARSFELD and P. RAO:
Linkage of general curves of large degree .................... 267
P. MAROSCIA:
Some problems and results on finite sets of
points in pn ................................................. 290
V.B. MEHTA and A. RAMANATHAN:
Homogeneous bun~es in characteristic p ....................... 315
I. MORRISON and U. PERSSON:
The group of sections on a rational elliptic surface ......... 321
D. MUMFORD:
On the Kodaira dimension of the Siegel
modular variety .............................................. 348
F. ORECCHIA:
Generalized Hilbert functions of Cohen-Macaulay varieties .... 376
L. ROBBIANO and G. VALLA:
Some curves in p3 are set-theoretic complete intersections .... 391
E. STAGNARO:
Constructing Enriques surfaces from quintics in p3 ........... 400
K
G. VAN DER GEER:
Prym surfaces and a Siegel modular threefold ................. 404
NO NOITARENEGED FO PROJECTIVE SEVRUC
by E. BALLICO and Ph. ELLIA
In this paper we study the problem of embedded deformations
of projective curves. A typical example is as follows : given in
p n a smooth curve C of degree d and a line L intersecting
C at only one point ( L not tangent to C ) does there exists
a scheme T and a subscheme Z of P n T ' flat over T such
that the generic fiber Z t is smooth and the special fiber is
C U L ? In the affirmative case we will say that C U L is
smoothable.
If C U L is smoothable then it is in Z(g, d+1 ; n) which
denotes the closure in Hilb pn of the set of nonsingular cur-
ves of genus g and degree d . By the way not every curve is
smoothable, for example (see IV.2) there exists a smooth curve C
of genus 9 and degree 8 , a line L intersecting C at only
one point such that C U L has no smooth embedded deformation.
On the other hand it follows from our method that if 0C(I) is
non special then C U L is in Z(g, d+1 ; n) In fact we prove
a stronger result by constructing the degeneration in small sub-
schemes of Z(g, d+1; n) (see Thm 111.5) In the same way let
C be a given curve of degree d in ~n , ~k a linear sub-
space of ~n and denote by Prd(C, ~k ) the closure in Hilb ~k
of the set of general projections of C on ~k Then we are
able to describe some reducible curves of Prd(C, ~k ) (see II)
Indeed the main idea of this paper is as follows : take a
curve Y in p n+1 , a line D intersecting Y at one point and an hyperpla-
ne H of ~n+| . Consider the flat family of curves in H obtained by projec-
ting Y from the points of D : the generic fiber will be a smooth curve in
Pr(Y, H) and the special fiber (over the point Y N D ) will be a reducible
curve C U L (see 1.1) . This is our starting point and all our results are
just variations on this theme.
Proposition 111.4 and Theorem 111.5 are largely used in B - E . Fi-
nally we plane in a future paper to use the results of § II to solve Hartshor-
ne's conjecture about generic projections of elllptic curves (see Ha,2 4.3.4)
NOTE : As this paper was finished we learned that Theorem 111.5. had been pro-
ved by Tannenbaum with a different method ( T2 ) .
NOTATIONS We work over an algebraically closed field K with ch(K) = O .
Two subschemes X and Y of ~n are said to be (quasi)-transversal at
x £ X n Y if they are nonsingular at x and if the natural map :
T x X ~ Tx Y ~--- Tx ~n is surj ective (resp of maximal rank) .
If C is a smooth curve of degree d in ~n and H is a linear subspace of
~n we denote by Prd(C ; H) the closure in Hilb H of the set of general pro-
jections of C in H . Note that Prd(C; H) is irreducible. If X is a varie-
ty and F an OX- module we put : F := HOmox(F , 0 X) , h°(F) := dim H°(X, F).
If X is a smooth curve, D a divisor on X and L a line bundle on X we
write : L(D) := L ~ O(D) . We denote by ~X the canonical sheaf on X.
x0
I . THE METHOD
In this section we prove Prop. 1.1 which is our main tool to cons-
truct embedded deformations of curves of type C U L . In the next sections
we will get our results from little variations of Prop. 1.1. Let us describe
the typical situation we will consider.
In ~n+1 (n > 2) take a non degenerate smooth curve C
of degree d and genus g , a point x of C and an hyperplane
H such that : x ~ H . We denote by ~ : pn+1 .... ~ H
n
the projection from x . Put Y := w (C) . We assume Y is non-
X
singular (of degree d-1 ) and we denote by y the point of Y
which is the image of x under ~ (i.e. : {y} = T C ~ H)
x X
In this situation we have :
1.1. PROPOSITION Let L be a line in H intersecting Y
only at y quasi-transversally, then Y U L is in Prd(L ; H)
PROOF Let D be a line intersecting C only at x , not
tangent to C , and intersecting L in a point t distinct from
y (see Fig I) . Now consider the projections of C from the
points of D \ {x} . This yields a family of curves, X CH D \{x}
parametrized by D \ {x} Since C is non degenerate we can
choose D such that, disregarding if necessary a finite number of
points, the fibers of X are smooth irreducible curves of degree
d and genus g . This family is flat ( Ha , II.9.9)
Furthermore there exists a unique subscheme X of H D ,
flat over D , whose restriction to HD \ {x} is X ( Ha II 9.8).
We claim that the fiber of X over x is Y U L
Obviously the fiber contains Y and, since the degree is preserved
by flatness, it also contains a line. This line is precisely L be-
cause every fiber has the double point ~ at x (see Fig I), in-
deed ~ arises from the projection of the tangent T x C . Now
from the exact sequence : o --* Oy UL --* OY ~ OL --+ OFnL ~--- o
and by the assumption : Y ~ L = {y} it follows that
pa(Y U L) = g , therefore the set theoretic fiber Y U L agrees
with the schematic fiber
X