Sui divisori di prima specie di una varietà algebrica non singolare

Palleschi, Marino

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 64 (1978) no. 4, pp. 367-373

Résumé

In a complex projective space we consider a non-singular algebraic variety $V_{d}$ of dimension $d \ge 2$. $|X|$ denotes a complete ample linear system on $V_{d}$ and $X_{m_{1}}, X_{m_{2}}, \cdots ,X_{m_{q}}$ denote $q \le d—1$ 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 $p \le q$ 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.

Bibliographie

[1] W. V. D. Hodge (1955) - A note on the Riemann-Roch Theorem, «Journal London Math. Soc.», 30, 291-296. | DOI | MR | Zbl

[2] K. Kodaira (1953) - On cohomology groups of compact analytic varieties with coefficients in some analytic faisceaux, «Proc. Nat. Acad. Sc. U.S.A.», 39, 865-868. | MR | Zbl

[3] K. Kodaira (1954) - Some results in the transcendental theory of algebraic varieties, «Annals of Math.», 59, 86-133. | DOI | MR | Zbl

[4] K. Kodaira e D. C. Spencer (1953) - Divisor class groups on algebraic varieties, «Proc. Nat. Acad. Sc. U.S.A.». 39. 872-877. | MR | Zbl

[5] E. Marchionna (1961) - Sui multipli del sistema delle sezioni iperpiane di una varietà algebrica non singolare, «Annali di Matem.». ser. IV, 54, 159-199. | DOI | MR | Zbl

[6] E. Marchionna (1962) - Sui multipli dei sistemi lineari d'ipersuperficie appartenenti ad una varietà algebrica pluriregolare, «Rendiconti di Matematica», (3-4), 21, 322-353. | MR | Zbl

[7] E. Marchionna (1971) - Sui divisori di prima specie di una varietà algebrica, «Symposia Mathematica» (Ist. Naz. Alta Matematica), vol. V, Academic Press, 439-456. | MR | Zbl

[8] J. P. Serre (1955) - Un théorème de dualité, «Comment. Math. Helv.», 29, 9-26. | fulltext EuDML | DOI | MR | Zbl

[9] J. P. Serre (1955) - Faisceaux algébriques cohérents, «Annals of Mathem.», 61, 197-278. | DOI | MR