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**On a problem of Zariski on dimensions of linear systems.**
*(English)*
Zbl 0822.14006

Let \(X\) be a nonsingular projective variety over the algebraically closed field \(k\), \({\mathcal L}\) a line bundle on \(X\) and \(h^ 0 ({\mathcal L}^{\otimes n}) = \dim_ k H^ 0(X, {\mathcal L}^{\otimes n})\). The Riemann-Roch problem consists in studying the function \(h^ 0({\mathcal L}^{\otimes n})\) for large \(n\). The problem was solved for curves and was first studied by Italian geometers for surfaces. O. Zariski [Ann. Math., II. Ser. 76, 560-615 (1962; Zbl 0124.370)] proved that if \(S\) is a nonsingular surface and \(D\) an effective divisor on \(S\) then \(h^ 0 ({\mathcal O}_ S (nD)) = P(n) + \Lambda (n)\) for \(n \gg 0\), where \(P(x)\) is a quadratic polynomial and \(\Lambda (n)\) is a bounded function. In this fundamental paper Zariski also stated that “it is an open question whether \(\Lambda (n)\) is always a periodic function of \(n\)”. Zariski himself obtained some partial results in this area. He proved that if \(D\) is an effective divisor on \(S\) such that the Kodaira dimension \(K(D) = \text{tr deg} (\bigoplus_{n \geq 0} H^ 0({\mathcal O}_ S (nD))) - 1\) is not greater than 1 then \(\bigoplus_{n \geq 0} H^ 0 ({\mathcal O}_ S (nD))\) is a finitely generated \(k\)-algebra and hence \(\Lambda (n)\) is periodic. Because \(K(D) \leq 2\) on a surface, the remaining case is \(K(D) = 2\).

The authors of the present paper solve the Zariski problem completely. The main results are the following:

Theorem 2: If \(\text{char} k = 0\), \(S\) is a normal surface proper over \(k\) and \(D\) is an effective Cartier divisor on \(S\), then for \(n \gg 0\), \(h^ 0 ({\mathcal O}_ S (nD)) = P(n) + \Lambda (n)\) where \(P(n)\) is a quadratic polynomial and \(\Lambda (n)\) is a periodic function.

Theorem 3: If \(k\) is a finite field, under the same hypothesis on \(S\), then \(\bigoplus_{n \geq 0} H^ 0 (S, {\mathcal O}_ S (nD))\) is a finitely generated \(k\)-algebra and \(h^ 0 ({\mathcal O}_ S (nD)) = P(n) + \Lambda (n)\), where \(P(n)\) is a quadratic polynomial and \(\Lambda (n)\) is periodic for \(n \gg 0\).

Theorem 3 is the stronger result because Zariski showed that \(\bigoplus_{n \geq 0} H^ 0({\mathcal O}_ s (nD))\) can be a nonfinitely generated \(k\)-algebra. The paper also contains many interesting other results. It is furnished a very nice but sophisticated example of a ruled surface \(X = \mathbb{P} ({\mathcal E})\), \(X @>\pi>>C\), \({\mathcal E} = {\mathcal O}_ \mathbb{C} + {\mathcal O}_ \mathbb{C} (P)\), over a field of positive characteristic \(p\) on which the Zariski problem has a negative answer for a line bundle \({\mathcal L} = {\mathcal O}_ X(1) \otimes \pi^* {\mathcal M}\). The Riemann-Roch problem in higher dimensions is also considered and it is given an example of an effective divisor \(D\) on a 3-fold \(X\) (projective and nonsingular) such that \(h^ 0 ({\mathcal O}_ X (nD))\) is a polynomial of degree 3 in \([n(2 - \sqrt 3/3)]\). In such way Zariski’s problem has a negative answer even for effective divisors in dimension greater than 2.

The authors of the present paper solve the Zariski problem completely. The main results are the following:

Theorem 2: If \(\text{char} k = 0\), \(S\) is a normal surface proper over \(k\) and \(D\) is an effective Cartier divisor on \(S\), then for \(n \gg 0\), \(h^ 0 ({\mathcal O}_ S (nD)) = P(n) + \Lambda (n)\) where \(P(n)\) is a quadratic polynomial and \(\Lambda (n)\) is a periodic function.

Theorem 3: If \(k\) is a finite field, under the same hypothesis on \(S\), then \(\bigoplus_{n \geq 0} H^ 0 (S, {\mathcal O}_ S (nD))\) is a finitely generated \(k\)-algebra and \(h^ 0 ({\mathcal O}_ S (nD)) = P(n) + \Lambda (n)\), where \(P(n)\) is a quadratic polynomial and \(\Lambda (n)\) is periodic for \(n \gg 0\).

Theorem 3 is the stronger result because Zariski showed that \(\bigoplus_{n \geq 0} H^ 0({\mathcal O}_ s (nD))\) can be a nonfinitely generated \(k\)-algebra. The paper also contains many interesting other results. It is furnished a very nice but sophisticated example of a ruled surface \(X = \mathbb{P} ({\mathcal E})\), \(X @>\pi>>C\), \({\mathcal E} = {\mathcal O}_ \mathbb{C} + {\mathcal O}_ \mathbb{C} (P)\), over a field of positive characteristic \(p\) on which the Zariski problem has a negative answer for a line bundle \({\mathcal L} = {\mathcal O}_ X(1) \otimes \pi^* {\mathcal M}\). The Riemann-Roch problem in higher dimensions is also considered and it is given an example of an effective divisor \(D\) on a 3-fold \(X\) (projective and nonsingular) such that \(h^ 0 ({\mathcal O}_ X (nD))\) is a polynomial of degree 3 in \([n(2 - \sqrt 3/3)]\). In such way Zariski’s problem has a negative answer even for effective divisors in dimension greater than 2.

Reviewer: M.I.Becheanu (Bucureşti)