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Let X be a reduced closed subscheme in ${\mathbb{P}}^n$ and $${\pi}_q:X{\rightarrow}Y={\pi}_q(X){\subset}{\mathbb{P}}^{n-1}$$ be an isomorphic projection from the center $q{\in}{\mathbb{P}}^n{\backslash}X$. Suppose that the minimal free presentation of $I_X$ is of the following form $$R(-3)^{{\beta}2,1}{\oplus}R(-4){\rightarrow}R(-2)^{{\beta}1,1}{\rightarrow}I_X{\rightarrow}0$$. In this paper, we prove that $H^1(I_X(k))=H^1(I_Y(k))$ for all $k{\geq}3$. This implies that Y is k-normal if and only if X is k-normal for $k{\geq}3$. Moreover, we also prove that reg(Y) ${\leq}$ max{reg(X), 4} and that $I_Y$ is generated by homogeneous polynomials of degree ${\leq}4$.
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참고문헌 (8)
참고문헌 신청
J. Ahn / 2011 / Graded mapping cone theorem, multisecants and syzygies / Journal of Algebra 331:243~262
J. Ahn / 2015 / On syzygies, degree, and geometric properties of projec-tive schemes with property N3,p / Journal of Pure and Applied Algebra 219:2724~2739
A. Alzati / 2002 / On the k-normality of projected algebraic varieties / Bull. Braz. Math. Soc. (N. S.) 33(1):27~48
D. Eisenbud / 2005 / Restriction linear syzygies: algebra and geometry / Compositio Math 141:1460~1478
D. Eisenbud / 2006 / Small schemes and varieties of minimal degree / Amer. J. Math 128(6):1363~1389
J. Ahn / 2015 / On syzygies, degree, and geometric properties of projec-tive schemes with property N3,p / Journal of Pure and Applied Algebra 219:2724~2739
A. Alzati / 2002 / On the k-normality of projected algebraic varieties / Bull. Braz. Math. Soc. (N. S.) 33(1):27~48
D. Eisenbud / 2005 / Restriction linear syzygies: algebra and geometry / Compositio Math 141:1460~1478
D. Eisenbud / 2006 / Small schemes and varieties of minimal degree / Amer. J. Math 128(6):1363~1389
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