Result: Pic^0 is always trivial, and Pic = NS is always free abelian, with rank increasing by 1 each time you blow up. For surfaces in $\mathbb {P}^3$ of degree at most 3 the calculation of $Pic …

2021年6月13日 · If $X$ is a non singular plane curve, what is the relationship between the Jacobian variety of $X$ $J(X)$ and the Picard group of divisors of degree 0 i.e. Pic $^0(X)$? I …

2018年8月14日 · $\mathrm{Pic}^0$ is meaningful only for complete varieties. If the variety is not normal, this is defined as an abelian group, but not as a scheme (see Variete du Picard, by CS …

By definition, $L$ is in $Pic^0$ if the homomorphism $\phi_L$ is identically zero, where $\phi: X \rightarrow Pic(X)$ is defined by sending $x$ to the isomorphism class of $T_x^*L \otimes L^{ …

2021年11月16日 · The set $\operatorname{Pic}^0(M) := H^1(M,\mathcal{O}_M)/Im(H^1(M,\mathbb{Z}))$ by definition. The $j$ in the exact sequence is …

2015年12月20日 · Let $Pic^0(X)$ denote the subgroup of $Pic(X)$ consisting of divisors of degree $0$. Let $n\in \mathbb{Z}$ be non-zero and $Pic^0(X)[n]$ denote the kernel of the …

0 → Pic(T) → Pic(XT) → PicX/S(T) is an exact sequence for all T. Proof. We may replace S by T and X by XT and assume that S = T to simplify the notation. Let N be an invertible OS …

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2015年9月22日 · Over $R$, the group scheme $Pic^{0,l}$ descends to the Néron model yielding a new geometric interpretation of its points and new combinatorial interpretations of the …

In Huybrechts' Complex Geometry: An Introduction, the Jacobian of a complex manifold X X, denoted Pic0(X) Pic 0 (X), is defined as the kernel of the map c1: Pic(X) → H2(X,Z) c 1: Pic …