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.