# Sui divisori di prima specie di una varietà algebrica non singolare

• 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.

## How to cite

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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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1. HODGE, W. V. D. (1955) - A note on the Riemann-Roch Theorem, «Journal London Math. Soc.», 30, 291-296. Zbl0065.14101MR78010DOI10.1112/jlms/s1-30.3.291
2. KODAIRA, K. (1953) - On cohomology groups of compact analytic varieties with coefficients in some analytic faisceaux, «Proc. Nat. Acad. Sc. U.S.A.», 39, 865-868. Zbl0051.14502MR63120
3. KODAIRA, K. (1954) - Some results in the transcendental theory of algebraic varieties, «Annals of Math.», 59, 86-133. Zbl0059.14605MR66690DOI10.2307/1969834
4. KODAIRA, K. e SPENCER, D. C. (1953) - Divisor class groups on algebraic varieties, «Proc. Nat. Acad. Sc. U.S.A.». 39. 872-877. Zbl0051.14601MR63122
5. MARCHIONNA, E. (1961) - Sui multipli del sistema delle sezioni iperpiane di una varietà algebrica non singolare, «Annali di Matem.». ser. IV, 54, 159-199. Zbl0103.14201MR133722DOI10.1007/BF02415350
6. MARCHIONNA, E. (1962) - Sui multipli dei sistemi lineari d'ipersuperficie appartenenti ad una varietà algebrica pluriregolare, «Rendiconti di Matematica», (3-4), 21, 322-353. Zbl0113.36503MR149357
7. MARCHIONNA, E. (1971) - Sui divisori di prima specie di una varietà algebrica, «Symposia Mathematica» (Ist. Naz. Alta Matematica), vol. V, Academic Press, 439-456. Zbl0212.53803MR276235
8. SERRE, J. P. (1955) - Un théorème de dualité, «Comment. Math. Helv.», 29, 9-26. Zbl0067.16101MR67489DOI10.1007/BF02564268
9. SERRE, J. P. (1955) - Faisceaux algébriques cohérents, «Annals of Mathem.», 61, 197-278. MR68874DOI10.2307/1969915

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