Feb 13, 2019 · A coarser way of trying to detect if a bundle is trivial is via characteristic classes: given a classifying map $f\colon X \to BG$ and a cohomology theory $h$ we can study the map $f^*\colon h^*(BG) \to h^*(X)$, and in particular try to determine weather it is the $0$ map or not.

Our goal now is to study the representability properties of the Picard functor. Recall the definition of the Picard group. Definition 1. Let X be a scheme. The Picard group Pic(X) is the set of line bundles (or invertible sheaves) on X with group operation given by tensor product. Recall the following well known fact. Lemma 1.

CURVES OF GIVEN p-RANK WITH TRIVIAL AUTOMORPHISM GROUP JEFFREY D. ACHTER, DARREN GLASS, AND RACHEL PRIES Abstract. Let k be an algebraically closed eld of characteristic p > 0. Suppose g 3 and 0 f g. We prove there is a smooth projective k-curve of genus g and p-rank f with no non-trivial automorphisms.

If g is one, then C is an elliptic curve, and K C is the trivial bundle. The global sections of the trivial bundle form a one-dimensional vector space, so the n -canonical map for any n is the map to a point.

Sample answers to your first question: If $S$ is Spec of a local ring, or of a UFD, then $H^1(S, \mathcal O_S^{\times})$ is trivial. The answer to your second question is no : $X = $Spec $\mathbb C[x,y]/(y^2 - x^3 +x) \to S =$ Spec $\mathbb C[x]$ gives a counterexample of a geometric nature, and $X =$Spec $\mathbb Z[\sqrt{-5}] \to S =$ Spec ...

An arbitrary automorphism can be realized as monodromy over the base a nodal curve by taking the trivial family over the normalization and gluing by the automorphism. But you may consider a nodal curve pathological.

smooth projective connected curve, such that p:(2) is trivial for every invertible sheaf 2 on S for which 2 ~ ... ~ 2 (n summands) is trivial. This is seen as follows: Let 2 be such a sheaf. Then the étale covering p : T - S constructed in the proof of the theorem is an abelian Galois covering of S with Galois group n03BC(k).

Let $X$ be a variety and $\mathbb{C}$ be the field of complex. Then $L = X \times \mathbb{C}$ is a trivial line bundle. The set of sections of this line bundle is $\Gamma(X, L)$ which consisting all functions $s: X \to L$ such that $\pi \cdot s = id$, where $\pi: L \to X$ is the projection map.

Sep 27, 2016 · We call the curve trivial if , and non-trivial otherwise. We refer to the complex numbers as the initial point and terminal point (or final point ) of the curve respectively, and refer to these two points collectively as the endpoints of the curve.

On an integral, proper scheme X over k a line bundle L is trivial if and only if both L and its dual have a non-zero global section. This can be checked after an arbitrary field extension. The proof of this criterion uses that the global section of O_X are a field.