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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 …

I work on complex numbers. Let $Q \subset \mathbb P^4$ be a quartic threefold that contains a double line $l$. I want to compute the normal sheaf of $l$ in $Q$, and I seem to find only two possible ca …

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 …

Suppose I have a morphism $f:X\to Y$ which is a GIT quotient of $X$ with respect to some reductive, linear group. Does the semistable $X^{ss}$ and stable locus $X^s\subset X$ determine completely the …

Due to difficulties in finding references I am not sure of what relative minimality is about for conic bundles. Suppose we're working on con. bun. over surfaces. The definition is ok: the fiber over …

I am looking for a proof/reference of the following simple fact, which I think it holds true. Let $S\subset \mathbb{P}^n$ be a surface embedded by a very ample linear system. Then I know that the dua …

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 …

Suppose I have a singular projective variety $X\subset \mathbb{P}^n$ that is reducible with $X=\bigcup_i X_i$ smooth irreducible components. That is, the irreducible components are smooth but $X$ is …