Abstract:
Let $k$ be a field with char $k \not= 2$, $X$ be an affine surface defined by the equation $z^2=P(x)y^2+Q(x)$ where $P(x), Q(x) \in k[x]$ are separable polynomials. We will investigate the rationality problem of $X$ in terms of the polynomials $P(x)$ and $Q(x)$. The necessary and sufficient condition is $s \leq 3$ with minor exceptions, where $s=s_1+s_2+s_3+s_4$, $s_1$ (resp. $s_2$, resp. $s_3$) being the number of $c \in \overline{k}$ such that $P(c)=0$ and $Q(c) \not\in k(c)^2$ (resp. $Q(c)=0$ and $P(c) \not\in k(c)^2$, resp. $P(c)=Q(c)=0$ and $-\frac{Q}{P}(c) \not\in k(c)^2$). $s_4=0$ or $1$ according to the behavior at $x=\infty$. $X$ is a conic bundle over $\mathbb{P}_k^1$, whose rationality was studied by Iskovskikh. Iskovskikh formulated his results in geometric language. This paper aims to give an algebraic counterpart.

Abstract:
We show that a standard conic bundle over a minimal rational surface is rational and its Jacobian splits as the direct sum of Jacobians of curves if and only if its derived category admits a semiorthogonal decomposition by exceptional objects and the derived categories of those curves. Moreover, such a decomposition gives the splitting of the intermediate Jacobian also when the surface is not minimal.

Abstract:
In this paper, it is proved that certain stable rank-3 vector bundles can be written as extensions of line bundles and stable rank-2 bundles. As an application, we show the rationality of certain moduli spaces of stable rank-3 bundles over the projective plane P^2.

Abstract:
A $\mathbb Q$-conic bundle is a proper morphism from a threefold with only terminal singularities to a normal surface such that fibers are connected and the anti-canonical divisor is relatively ample. We study the structure of $\mathbb Q$-conic bundles near their singular fibers. One corollary to our main results is that the base surface of every $\mathbb Q$-conic bundle has only Du Val singularities of type A (a positive solution of a conjecture by Iskovskikh). We obtain the complete classification of $\mathbb Q$-conic bundles under the additional assumption that the singular fiber is irreducible and the base surface is singular.

Abstract:
We discuss natural connections between three objects: quadratic forms with values in line bundles, conic bundles and quaternion orders. We use the even Clifford algebra, and the Brauer-Severi Variety, and other constructions to give natural bijections between these objects under appropriate hypothesis. We then restrict to a surface base and we express the second Chern class of the order in terms $K^3$ and other invariants of the corresponding conic bundle. We find the conic bundles corresponding to minimal del Pezzo quaterion orders and we discuss problems concerning their moduli.

Abstract:
For conic bundles on a smooth variety (over a field of characteristic $\ne 2$) which degenerate into pairs of distinct lines over geometric points of a smooth divisor, we prove a theorem which relates the Brauer class of the non-degenerate conic on the complement of the divisor to the covering class (Kummer class) of the 2-sheeted cover of the divisor defined by the degenerate conic, via the Gysin homomorphism in etale cohomology. This theorem is the algebro-geometric analogue of a topological result proved earlier.

Abstract:
Let k be an arbitrary field of characteristic zero. In this paper we study quotients of k-rational conic bundles over projective line by finite groups of automorphisms. We construct smooth minimal models for such quotients. We show that any quotient is birationally equivalent to a quotient of other k-rational conic bundle cyclic group of order $2^k$, dihedral group of order $2^k$, alternating group of degree $4$, symmetric group of degree $4$ or alternating group of degree $5$ effectively acting on the base of conic bundle. Also we construct infinitely many examples of such quotients which are not k-birationally equivalent to each other.

Abstract:
Roughly speaking, a conic bundle is a surface, fibered over a curve, such that the fibers are conics (not necessarily smooth). We define stability for conic bundles and construct a moduli space. We prove that (after fixing some invariants) these moduli spaces are irreducible (under some conditions). Conic bundles can be thought of as generalizations of orthogonal bundles on curves. We show that in this particular case our definition of stability agrees with the definition of stability for orthogonal bundles. Finally, in an appendix by I. Mundet i Riera, a Hitchin-Kobayashi correspondence is stated for conic bundles.

Abstract:
Let the threefold X be a general smooth conic bundle over the projective plane P(2), and let (J(X), Theta) be the intermediate jacobian of X. In this paper we prove the existence of two natural families C(+) and C(-) of curves on X, such that the Abel-Jacobi map F sends one of these families onto a copy of the theta divisor (Theta), and the other -- onto the jacobian J(X). The general curve C of any of these two families is a section of the conic bundle projection, and our approach relates such C to a maximal subbundle of a rank 2 vector bundle E(C) on C, or -- to a minimal section of the ruled surface P(E(C)). The families C(+) and C(-) correspond to the two possible types of versal deformations of ruled surfaces over curves of fixed genus g(C). As an application, we find parameterizations of J(X) and (Theta) for certain classes of Fano threefolds, and study the sets Sing(Theta) of the singularities of (Theta).

Abstract:
We give a method to construct stable vector bundles whose rank divides the degree over curves of genus bigger than one. The method complements the one given by Newstead. Finally, we make some systematic remarks and observations in connection with rationality of moduli spaces of stable vector bundles.