Sui divisori di prima specie di una varietà algebrica non singolare

Marino Palleschi

Sui divisori di prima specie di una varietà algebrica non singolare

Marino Palleschi

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti (1978)

Volume: 64, Issue: 4, page 367-373
ISSN: 0392-7881

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Abstract

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In a complex projective space we consider a non-singular algebraic variety

V_{d}

of dimension

d\geq 2

.

|X|

denotes a complete ample linear system on

V_{d}

and

X_{m_{1}},X_{m_{2}},\cdots,X_{m_{q}}

denote

q\leq d—1

non-singular hypersurfaces belonging to

q

positive multiples

|m_{1}X|,\cdots,|m_{q}X|

of the linear system

|X|

. We suppose every subvariety

V_{d-i}=\displaystyle\bigcap^{i}_{j=1}X_{m_{j}}

(i=1,2,\cdots,q)

is non singular and has a regular dimension

d — i

. In this case the subvariety

V_{d-q}=\displaystyle\bigcap^{q}_{j=1}X_{m_{j}}

is called a quasi-characteristic variety of index

q

of the system

|X|

. A divisor

A

of

V_{d}

is said to be

q

-times of the first kind mod

|X|

if for each relative integer

l

the complete linear system

|lX—A|

, belonging to

V_{d}

, cuts out a complete system on every quasi-characteristic variety

V_{d-q}

of the system

|X|

. The above conditions can be reduced. In fact if for each

l

the complete system

|lX—A|

cuts out a complete system on a fixed quasi-characteristic variety of index

q

of the system

|X|

, then the complete system

|lX-A|

cuts out a complete system on any quasi-characteristic variety of index

p\leq q

of the system

|X|

. We denote with

H^{q}(V_{d},\mathcal{O}(D))

the

q

-th cohomology module of

V_{d}

with coefficients in the sheaf

\mathcal{O}(D)

of germs of meromorphic functions which are multiples of the divisor

— D

. With the previous notations, a characteristic condition for

A

to be

q

-times of the first kind mod

|X|

is that

H^{p}(V_{d},\mathcal{(O)}(lX-A))=0

, for each integer

l,p=1,2,\cdots,q

. A characteristic condition for

A

to be

q

-times of the first kind mod a suitable multiple of every ample linear system is that

H^{p}(V_{d},\mathcal{O}(-A))=(0)

,

(p=1,2,\cdots,q)

. We recall that the theory of divisors of the first kind was introduced and developed with geometrical language and instruments by Marchionna (cfr. [6], [7]). In this paper (and in the following Note II with the same title) we reconstruct the whole theory in an independent way, by employing cohomology theory.

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Palleschi, Marino. "Sui divisori di prima specie di una varietà algebrica non singolare." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti 64.4 (1978): 367-373. <http://eudml.org/doc/290060>.

@article{Palleschi1978,
author = {Palleschi, Marino},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti},
language = {ita},
month = {4},
number = {4},
pages = {367-373},
publisher = {Accademia Nazionale dei Lincei},
title = {Sui divisori di prima specie di una varietà algebrica non singolare},
url = {http://eudml.org/doc/290060},
volume = {64},
year = {1978},
}

TY - JOUR
AU - Palleschi, Marino
TI - Sui divisori di prima specie di una varietà algebrica non singolare
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
DA - 1978/4//
PB - Accademia Nazionale dei Lincei
VL - 64
IS - 4
SP - 367
EP - 373
LA - ita
UR - http://eudml.org/doc/290060
ER -

References

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MARCHIONNA, E. (1962) - Sui multipli dei sistemi lineari d'ipersuperficie appartenenti ad una varietà algebrica pluriregolare, «Rendiconti di Matematica», (3-4), 21, 322-353. Zbl0113.36503 MR149357
MARCHIONNA, E. (1971) - Sui divisori di prima specie di una varietà algebrica, «Symposia Mathematica» (Ist. Naz. Alta Matematica), vol. V, Academic Press, 439-456. Zbl0212.53803 MR276235
SERRE, J. P. (1955) - Un théorème de dualité, «Comment. Math. Helv.», 29, 9-26. Zbl0067.16101 MR67489 DOI10.1007/BF02564268
SERRE, J. P. (1955) - Faisceaux algébriques cohérents, «Annals of Mathem.», 61, 197-278. MR68874 DOI10.2307/1969915

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