Some remarks on the Kp,1$\mathcal {K}_{p,1}$ theorem.
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| Title: | Some remarks on the Kp,1$\mathcal {K}_{p,1}$ theorem. |
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| Authors: | Kim, Yeongrak1,2 (AUTHOR), Moon, Hyunsuk3 (AUTHOR) hsmoon87@kias.re.kr, Park, Euisung4 (AUTHOR) |
| Source: | Mathematische Nachrichten. Sep2024, Vol. 297 Issue 9, p3531-3545. 15p. |
| Subject Terms: | *PICARD number, *BETTI numbers |
| Abstract: | Let X$X$ be a non‐degenerate projective irreducible variety of dimension n≥1$n \ge 1$, degree d$d$, and codimension e≥2$e \ge 2$ over an algebraically closed field K$\mathbb {K}$ of characteristic 0. Let βp,q(X)$\beta _{p,q} (X)$ be the (p,q)$(p,q)$th graded Betti number of X$X$. Green proved the celebrating Kp,1$\mathcal {K}_{p,1}$‐theorem about the vanishing of βp,1(X)$\beta _{p,1} (X)$ for high values for p$p$ and potential examples of nonvanishing graded Betti numbers. Later, Nagel–Pitteloud and Brodmann–Schenzel classified varieties with nonvanishing βe−1,1(X)$\beta _{e-1,1}(X)$. It is clear that βe−1,1(X)≠0$\beta _{e-1,1}(X) \ne 0$ when there is an (n+1)$(n+1)$‐dimensional variety of minimal degree containing X$X$, however, this is not always the case as seen in the example of the triple Veronese surface in P9$\mathbb {P}^9$. In this paper, we completely classify varieties X$X$ with nonvanishing βe−1,1(X)≠0$\beta _{e-1,1}(X) \ne 0$ such that X$X$ does not lie on an (n+1)$(n+1)$‐dimensional variety of minimal degree. They are exactly cones over smooth del Pezzo varieties, whose Picard number is ≤n−1$\le n-1$. [ABSTRACT FROM AUTHOR] |
| Database: | Academic Search Index |
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Nájsť tento článok vo Web of Science | Abstract: | Let X$X$ be a non‐degenerate projective irreducible variety of dimension n≥1$n \ge 1$, degree d$d$, and codimension e≥2$e \ge 2$ over an algebraically closed field K$\mathbb {K}$ of characteristic 0. Let βp,q(X)$\beta _{p,q} (X)$ be the (p,q)$(p,q)$th graded Betti number of X$X$. Green proved the celebrating Kp,1$\mathcal {K}_{p,1}$‐theorem about the vanishing of βp,1(X)$\beta _{p,1} (X)$ for high values for p$p$ and potential examples of nonvanishing graded Betti numbers. Later, Nagel–Pitteloud and Brodmann–Schenzel classified varieties with nonvanishing βe−1,1(X)$\beta _{e-1,1}(X)$. It is clear that βe−1,1(X)≠0$\beta _{e-1,1}(X) \ne 0$ when there is an (n+1)$(n+1)$‐dimensional variety of minimal degree containing X$X$, however, this is not always the case as seen in the example of the triple Veronese surface in P9$\mathbb {P}^9$. In this paper, we completely classify varieties X$X$ with nonvanishing βe−1,1(X)≠0$\beta _{e-1,1}(X) \ne 0$ such that X$X$ does not lie on an (n+1)$(n+1)$‐dimensional variety of minimal degree. They are exactly cones over smooth del Pezzo varieties, whose Picard number is ≤n−1$\le n-1$. [ABSTRACT FROM AUTHOR] |
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| ISSN: | 0025584X |
| DOI: | 10.1002/mana.202400004 |