Theorem of Enriques-Petri type for a very ample invertible sheaf on a curve of genus three

1983 ◽  
Vol 183 (3) ◽  
pp. 343-353 ◽  
Author(s):  
Masaaki Homma
Keyword(s):  
Invertible Sheaf ◽  
Ample Invertible Sheaf

scholarly journals The family of lines on the Fano threefold V5

1989 ◽  
Vol 116 ◽  
pp. 111-122 ◽  
Author(s):  
Mikio Furushima ◽  
Noboru Nakayama
Keyword(s):  
Linear System ◽  
Betti Number ◽  
Invertible Sheaf ◽  
The Family ◽  
Anticanonical Divisor ◽  
Fano Threefold ◽  
Ample Invertible Sheaf

A smooth projective algebraic 3-fold V over the field C is called a Fano 3-fold if the anticanonical divisor — Kv is ample. The integer g = g(V) = ½(- Kv)3 is called the genus of the Fano 3-fold V. The maximal integer r ≧ 1 such that ϑ(— Kv)≃ ℋ r for some (ample) invertible sheaf ℋ ε Pic V is called the index of the Fano 3-fold V. Let V be a Fano 3-fold of the index r = 2 and the genus g = 21 which has the second Betti number b2(V) = 1. Then V can be embedded in P6 with degree 5, by the linear system |ℋ|, where ϑ(— Kv)≃ ℋ2 (see Iskovskih [5]). We denote this Fano 3-fold V by V5.

Very Ample Linear Systems on Blowings-Up at General Points of Projective Spaces

2002 ◽  
Vol 45 (3) ◽  
pp. 349-354 ◽  
Author(s):  
Marc Coppens
Keyword(s):  
Projective Space ◽  
Linear Systems ◽  
Exceptional Divisor ◽  
Closed Field ◽  
Invertible Sheaf ◽  
Algebraically Closed Field ◽  
Structure Sheaf ◽  
Blowing Up ◽  
Dimensional Projective Space ◽  
Ample Invertible Sheaf

AbstractLet Pn be the n-dimensional projective space over some algebraically closed field k of characteristic 0. For an integer t ≥ 3 consider the invertible sheaf O(t) on Pn (Serre twist of the structure sheaf). Let , the dimension of the space of global sections of O(t), and let k be an integer satisfying 0 < k ≤ N − (2n + 2). Let P1,…,Pk be general points on Pn and let π : X → Pn be the blowing-up of Pn at those points. Let Ei = π−1(Pi) with 1 ≤ i ≤ k be the exceptional divisor. Then M = π*(O(t)) ⊗ OX(−E1 — … — Ek) is a very ample invertible sheaf on X.

A STRENGTHENING OF RESOLUTION OF SINGULARITIES IN CHARACTERISTIC ZERO

2003 ◽  
Vol 86 (2) ◽  
pp. 327-357 ◽  
Author(s):  
A. BRAVO ◽  
O. VILLAMAYOR U.
Keyword(s):  
Characteristic Zero ◽  
Resolution Of Singularities ◽  
Closed Subscheme ◽  
Subject Classification ◽  
Invertible Sheaf ◽  
Simple Manner ◽  
Birational Morphism ◽  
Exceptional Locus

Let $X$ be a closed subscheme embedded in a scheme $W$, smooth over a field ${\bf k}$ of characteristic zero, and let ${\mathcal I} (X)$ be the sheaf of ideals defining $X$. Assume that the set of regular points of $X$ is dense in $X$. We prove that there exists a proper, birational morphism, $\pi : W_r \longrightarrow W$, obtained as a composition of monoidal transformations, so that if $X_r \subset W_r$ denotes the strict transform of $X \subset W$ then:(1) the morphism $\pi : W_r \longrightarrow W$ is an embedded desingularization of $X$ (as in Hironaka's Theorem);(2) the total transform of ${\mathcal I} (X)$ in ${\mathcal O}_{W_r}$ factors as a product of an invertible sheaf of ideals ${\mathcal L}$ supported on the exceptional locus, and the sheaf of ideals defining the strict transform of $X$ (that is, ${\mathcal I}(X){\mathcal O}_{W_r} = {\mathcal L} \cdot {\mathcal I}(X_r)$).Thus (2) asserts that we can obtain, in a simple manner, the equations defining the desingularization of $X$.2000 Mathematical Subject Classification: 14E15.

scholarly journals Some remarks on the cohomology of an invertible sheaf

1979 ◽  
Vol 82 (4) ◽  
pp. 473-480
Author(s):  
Mauro Beltrametti ◽  
Paolo Francia
Keyword(s):  
Invertible Sheaf

Distribution of Weierstrass Points on Rational Cuspidal Curves

1990 ◽  
Vol 33 (2) ◽  
pp. 184-189
Author(s):  
John B. Little
Keyword(s):  
Explicit Formula ◽  
Theta Function ◽  
Rational Curve ◽  
Invertible Sheaf ◽  
Smooth Curves ◽  
Weierstrass Points ◽  
Analogous Question ◽  
Tensor Powers

AbstractWe study the set W(𝓛) of Weierstrass points of all positive tensor powers of an invertible sheaf 𝓛 on an irreducible rational curve X with g ≧ 2 ordinary cusps. Using an idea from B. Olsen's study of the analogous question on smooth curves, and an explicit formula for the "theta function" of a cuspidal rational curve, we show that W(𝓛) is never dense on X (in contrast to the case of smooth curves of genus g ≧ 2).

Some remarks on the cohomology of an invertible sheaf

1979 ◽  
Vol 82 (1) ◽  
pp. 473-480
Author(s):  
Mauro Beltrametti ◽  
Paolo Francia
Keyword(s):  
Invertible Sheaf
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