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This is a partial answer. There are many papers on nets of quadrics (see e.g. work of Debarre and Beauville). Here the discriminant locus is a plane curve. If the base-locus of the net is non-singula …

Depending on the interpretation of your question, the answer is Yes. In fact the Hirzebruch surface $\mathbb{F}_n =\mathbb{P}(\mathcal{O}\oplus \mathcal{O}(n))$ is the quotient of $X = \mathbb{A}^2 \ …

Bjorn Poonen and his coauthors have studied topics of this type in a series of papers on "Bertini Theorem over Finite Fields". See for example the paper: Poonen, Charles - Bertini irreducibility the …

Yes if $K=\mathbb{R}$ for example, but no in general. Namely this fails for curves of genus $1$, over $\mathbb{Q}$, say. Given an elliptic curve $E$ over $\mathbb{Q}$ and a positive integer $d$, a ge …

The corresponding projective surface $$S: xyz + (x + y + z)w^2=0 \subset \mathbb{P}^3,$$ is a singular cubic surface - singular cubic surfaces are special. It has three singularities, each of which h …

Let $A$ be a finitely generated $\mathbb{Z}$-algebra which is a UFD. Then (a special case of) the Bass conjecture states that $K_0(A)$ is a finitely generated abelian group. As far as I am aware, this …