## Projective normality of algebraic curves and its application to surfaces.(English)Zbl 1127.14028

Let $$C$$ be a smooth curve of genus $$g$$ and $$L \in \text{Pic}^d(C)$$. $$L$$ is said to be normally generated if it is very ample and the natural map $$S^m(H^0(C,L)) \to H^0(C,L^{\otimes m})$$ is surjective for all $$m>0$$. Here the authors prove that any very ample degree $$d$$ $$L$$ is normally generated if $$d > \max \{2g+2-4h^1(C,L),2g-(g-1)/6-2h^1(C,L)\}$$. Now assume the existence of a degree $$3$$ map $$u: C \to C'$$ with $$C'$$ of genus $$q$$. They prove that $$\omega _C(-u^\ast (D))$$ is very ample and normally generated if $$D$$ is an effective divisor of degree $$z$$ on $$C'$$ and $$4q < z < (g-1)/6 -2q\}$$. They also prove (using hyperplane sections) the projective normality of certain surfaces.

### MSC:

 14H51 Special divisors on curves (gonality, Brill-Noether theory) 14H50 Plane and space curves 14C20 Divisors, linear systems, invertible sheaves 14J99 Surfaces and higher-dimensional varieties
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### References:

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