论文标题
Frobenius非古典曲面
Frobenius nonclassical hypersurfaces
论文作者
论文摘要
如果有限字段$ \ mathbb {f} _q $在$ q $ - Q $ -th frobenius内态下的每个几何点的图像仍然处于独特的超平面切线，则在有限的字段上进行了平滑的高度表面。在本文中，我们在此类超曲面的程度上建立了尖锐的下限和上限，为达到最大程度的人提供了特征，在表面情况下证明，当他们的学位达到最小值时，它们是隐士。我们还证明，$ \ mathbb {f} _q $ - 合理点在frobenius nonclassical hyperface上形成了相对于线条的阻止集，这表明存在许多$ \ mathbb {f} _q $ - 点。
A smooth hypersurface over a finite field $\mathbb{F}_q$ is called Frobenius nonclassical if the image of every geometric point under the $q$-th Frobenius endomorphism remains in the unique hyperplane tangent to the point. In this paper, we establish sharp lower and upper bounds for the degrees of such hypersurfaces, give characterizations for those achieving the maximal degrees, and prove in the surface case that they are Hermitian when their degrees attain the minimum. We also prove that the set of $\mathbb{F}_q$-rational points on a Frobenius nonclassical hypersurface form a blocking set with respect to lines, which indicates the existence of many $\mathbb{F}_q$-points.
