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Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 41, No. 1, pp. 279-290 (2000)
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Koszul Cohomology and $k$-Normality of a Projective Variety

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A. Alzati, G. M. Besana

**Keywords:** Koszul cohomology, $k$-normality, scrolls \font\bb=msbm10 \font\tbb=msbm7 \font\eu=eusm10 \def\bP{\hbox{\bb P}} \def\tbP{\hbox{\tbb P}} \def\sO{\hbox{\eu O}} Let $X$ be a smooth projective variety and let $L$ be a very ample divisor of $X$ embedding it in $\bP^N$. In this paper we use the Koszul groups of $X$ to get information about the $k$-normality of $X$ (i.e. the surjectivity of the map $H^0(\bP^N,\sO_{\tbP^N}(k))\rightarrow H^0(X,kL)$ via an upper bound for the degree of the generators of $\oplus_{t\ge 0}H^0(X,tL)$. The above idea is applied to some scrolls over curves and surfaces and to some other varieties, by using also results due to Green and Butler.

**Classification (MSC2000):** 14J40

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