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Let $V$ be the Veronese surface, obtained as the image of $\mathbb{P}^2$ in $\mathbb{P}^5$ by the complete linear system of conics. I understand that $V$ can degenerate to the union of a cubic scroll …

It depends on each case. In general there is an explicit birational map from $X$ to a rational variety (typically $P^4$, or a quadric, as it is the case for pfaffian cubics), whose indeterminacy locus …

Let $S$ be a smooth algebraic variety, and suppose $X\to S$ is a smooth morphism of schemes such that the geometric fibers are all projective spaces. Let us suppose that the dimension of the fibers is …

Are there classes of algebraic varieties for which algebraic and rational equivalence for algebraic cycles coincide? (references also appreciated)

A motive over a field $k$ is of abelian type if it belongs to the thick and rigid subcategory of Chow motives spanned by the motives of abelian varieties over $k$. I understand that this is the smalle …

here https://arxiv.org/pdf/1409.5033.pdf in section 5 there's an example of unirational, non-rational, 3fold moduli space. I am not sure whether it is a conic bundle, though. Probably not.

Two K3 surfaces show up when talking about cubic fourfolds containing a plane. Let $P\subset X\subset \mathbb{P}^5$ be the plane inside the cubic. Since $P$ is cut out by 3 linear equations then $X$ h …

It is well known that any birational morphism between projective varieties is a sequence of blow ups. Suppose now that I have a morphism $f:X \to Y$ with positive dimensional fibers, that is a project …

By singular K3 I mean a smooth complex K3 with Néron-Severi rank equal to 20. Are singular K3 surfaces dense in the moduli space of polarized K3 surfaces?

By Hitchin fibration I mean the usual morphism from the coarse moduli space of semi-stable Higgs bundles to the Hitchin base (i.e. the direct sum of spaces of global sections of powers of the canonica …

What is the standard reference for a definition , valid in all characteristics, of the residue in a point of a rational differential form on a curve?

Suppose I have two smooth projective varieties $X$ and $Y$ in $\mathbb{P}^n$, that intersect along a smooth subvariety $Z$. Is there a formula to compute the degree of the join variety $J(X,Y)$ of $X$ …

Suppose we have a smooth subvariety $X\subset Gr(2,n)$ of a Grassmannian, that can be expressed as usual as a linear combination of Schubert cycles. I would like to obtain information on the canonical …

Let $L$ be a line bundle on a smooth algebraic variety $X$. Let $\sigma_i:U_i \times \mathbb{C} \to L_{|U_i} $ be its local trivializaations and $u_{ij}$ the transition functions satisfying $\sigma_j= …

Does anybody have a reference for invariants of configurations of linear subspaces in the projective space? In particular I would be curious to see an explicit expression of the invariant functions o …