# Questions tagged [projective-varieties]

In algebraic geometry, a projective variety over an algebraically closed field $k$ is a subset of some projective $n$-space $\mathbb P^n$ over $k$ that is the zero-locus of some finite family of homogeneous polynomials of $n + 1$ variables with coefficients in $k$, that generate a prime ideal, the defining ideal of the variety

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### Non-unique completion of a flat family of smooth projective varieties

Let $\mathbb{k}$ be an algebraically closed field of characteristic 0. Denote $S=\mathrm{Spec}\:\mathbb{k}[t]$, $U=\mathrm{Spec}\:\mathbb{k}[t, t^{-1}]$, $Z=\mathrm{Spec}\:\mathbb{k}[t]/(t)$. What is ...
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### Projective embeddings of quotients of normal varieties

Let $X$ be a normal complex projective variety of dimension $m$, $G$ be a finite subgroup of $\mathrm{Aut}(X)$, and $Y = X / G$ be the quotient. I am particularly interested in the case where $X$ is a ...
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### Closed points of a closed subscheme of $\mathbb{P}^n$ over the residue field and the fraction field of a valuation ring $R$

Let $(R, M)$ be a valuation ring with algebraically closed fraction field $k$. Let $L = R/M$ be the residue field of $R$; it follows that $L$ is algebraically closed. I would like to understand the ...
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### Chow form of closure of product of affine varieties given the chow forms of their closurs

This question is about the connection between $\overline{X\times Y}$ and $\overline{X}$,$\overline{Y}$ where $X\subset \mathbb{A}^{n},\;Y\subset\mathbb{A}^{m}$ are affine varities over an ...
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### Bertini theorem for connectedness

Let $X$ be a geometrically irreducible, possibly singular projective variety over an infinite field $k$. Assume that the dimension of $X$ is at least 2. Can there exist a hyperplane section of $X$ ...
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### When is the pushout of projective varieties along embeddings a projective variety?

From Karl Schwede's paper "Gluing schemes and a scheme without closed points'', I know that there exists a pushout of schemes for closed embeddings. Now, if I start in the projective world I would ...
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### Moduli space is a Calabi-Yau manifold?

I asked a question here where a moduli space of flat connection is related to the $n$-dimensional complex projective space: $$\Bbb E/S_n \cong \Bbb P^{n-1}.$$ This is related to a 4d SU(N) Yang-...
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### The isometry groups of flag manifolds

For any sequence of integers $0<n_1<...<n_k$, there is a flag manifold of type $(n_1, ..., n_k)$, which is the collection of ordered sets of vector subspaces of $R^{(n_k)}$ $(V_1, ..., V_k)$ ...
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### Complex projective algebraic variety, moduli space of flat connections, and instantons

In Looijenga's work below, if I understand correctly, it shows that Statement 1: At an algebraic variety, the moduli space of SU($N$) flat connections on a 2-torus $T^2$ is given by the space of ...
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### A non-rational variety with a full exceptional collection?

Does there exist a smooth non-rational projective variety whose bounded derived category of coherent sheaves admits a full exceptional collection? I could not find any examples in the literature (for ...
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Let $(A,g)$ be a graded commutative ring, where $A$ denotes the commutative ring, and $g$ its grading. What can be said about the set $\mathcal{G}_A := \{ \mathrm{Proj}\Big((A,g)\Big)\ \vert\ g \}$? ...
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### Fiber product of projective varieties and ample line bundles

Let $X, Y$ be smooth, projective varieties, $L_X$ and $L_Y$ are very ample line bundles on $X$ and $Y$, respectively. If I understand correctly, $\mbox{pr}_1^*L_X \otimes \mbox{pr}_2^*L_Y$ is a very ...
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### Relations between an projective variety and galois cohomology

Let $f_1, \cdots, f_k$ be homogeneous polynomials over $\mathbb{Q}[x_0, \cdots, x_n]$. They define an projective variety $X$ over $\mathbb{P}^n(\mathbb{C})$, namely their set of zeros X = Z(f_1, \...
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### Birational immersion of projective plane

Let $X$ be a smooth, projective $3$-fold. Under what condition on $X$, does there exist a birational immersion from the projective plane $\mathbb{P}^2$ to $X$?
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### Where can I find a proof of identity of $H^1(X,T_X)$ and a quotient by the jacobian?

I'm reading some notes on hodge theory by Charles Siegel which makes a claim on page 16 relating the space of deformations of a smooth projective hypersurface $X$ with the jacobian ideal. More ...