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1. CMB 2011 (vol 54 pp. 430)
Complete Families of Linearly Non-degenerate Rational Curves We prove that every complete family of linearly non-degenerate
rational curves of degree $e > 2$ in $\mathbb{P}^{n}$ has at most $n-1$
moduli. For $e = 2$ we prove that such a family has at most $n$
moduli. The general method involves exhibiting a map from the base of
a family $X$ to the Grassmannian of $e$-planes in $\mathbb{P}^{n}$ and
analyzing the resulting map on cohomology.
Categories:14N05, 14H10 |
2. CMB 2002 (vol 45 pp. 349)
Very Ample Linear Systems on Blowings-Up at General Points of Projective Spaces Let $\mathbf{P}^n$ be the $n$-dimensional projective space over some
algebraically closed field $k$ of characteristic $0$. For an integer
$t\geq 3$ consider the invertible sheaf $O(t)$ on $\mathbf{P}^n$ (Serre
twist of the structure sheaf). Let $N = \binom{t+n}{n}$, the
dimension of the space of global sections of $O(t)$, and let $k$ be an
integer satisfying $0\leq k\leq N - (2n+2)$. Let $P_1,\dots,P_k$
be general points on $\mathbf{P}^n$ and let $\pi \colon X \to
\mathbf{P}^n$ be the blowing-up of $\mathbf{P}^n$ at those points.
Let $E_i = \pi^{-1} (P_i)$ with $1\leq i\leq k$ be the exceptional
divisor. Then $M = \pi^* \bigl( O(t) \bigr) \otimes O_X (-E_1 -
\cdots -E_k)$ is a very ample invertible sheaf on $X$.
Keywords:blowing-up, projective space, very ample linear system, embeddings, Veronese map Categories:14E25, 14N05, 14N15 |