Jason Starr
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Registered User
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1d |
answered | analytically isomorphic singularities |
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May 21 |
accepted | Proving a variety is not unirational |
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May 21 |
answered | enumerative Gromov-Witten invariants |
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May 21 |
answered | Proving a variety is not unirational |
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May 20 |
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When is the intersection of an isolated normal singularity with a generic linear subspace through that singularity normal? E.g., affine cone over a general embedding of an Abelian surface. |
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May 19 |
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Proving a variety is not unirational This problem is still open. |
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May 18 |
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Connectedness of hyperplane sections (reference request) The Fulton-Hansen theory is more general, but this particular case is the Lemma of Enriques-Severi-Zariski, which is older. |
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May 17 |
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Vector bundles on Stein Manifolds These types of results are sometimes also called "Grauert's Oka principle". So perhaps there are earlier partial results of Oka (e.g., when $k$ equals $1$). |
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May 17 |
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There are many varieties with ample canonical bundle What makes you believe that there is an irreducible component of your moduli space of positive dimension? If you are asking whether or not that is true, I suggest editing your question a bit. |
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May 8 |
revised |
Cohomology of twisted holomorphic forms on Fano threefolds Amplified computation of normal bundle |
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May 8 |
revised |
Cohomology of twisted holomorphic forms on Fano threefolds Tried to correct LaTeX (not sure what is wrong) |
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May 8 |
revised |
Cohomology of twisted holomorphic forms on Fano threefolds Addressed OP's question. |
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May 7 |
answered | Cohomology of twisted holomorphic forms on Fano threefolds |
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May 7 |
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Shafarevich’s theorem for elliptic curves defined over function field of algebraic curve over algebraically closed field Oops, Moret-Bailly's examples are pencils of Abelian surfaces, not elliptic curves. |
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May 7 |
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Shafarevich’s theorem for elliptic curves defined over function field of algebraic curve over algebraically closed field In positive characteristic there are the Moret-Bailly pencils. |
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May 4 |
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Global sections of twisting ideal sheaf of a smooth closed point on a projective space You have incorrectly computed $h^0(\mathbb{P}^2,\mathcal{O}(1)/\mathcal{I}_x^3\mathcal{O}(1))$. The dimension should be $6$, not $3$. |
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May 3 |
answered | Uniqueness of the canonical etale coverings |
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May 3 |
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Global sections of twisting ideal sheaf of a smooth closed point on a projective space Did you try to compute this for yourself? |
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May 1 |
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are moduli stacks deligne-mumford stacks in general Correction: As userN points out, I missed the OP's hypothesis that the stack is defined over $\text{Spec}(\mathbb{C})$. For stacks over $\text{Spec}(\mathbb{C})$, I agree with Ravi: every algebraic stack with finite diagonal is Deligne-Mumford. This follows from Artin's theorems. |
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May 1 |
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are moduli stacks deligne-mumford stacks in general As nosr points out, this answer is incorrect. For a very natural example, the stacks of Kontsevich stable maps are not (usually) Deligne-Mumford in positive characteristic. |
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Apr 30 |
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Are all Equivariant Bundles of a Total Flag Manifold Constructable from Line Bundles? @Ago: An iterated extension is a $G$-equivariant vector bundle (or, equivalently, locally free sheaf) together with a $G$-invariant filtation by $G$-equivariant vector subbundles (locally free subsheaves with locally free quotient) whose associated subquotients are each $G$-equivariant line bundles (invertible sheaves). I am confused by your notation for $U(1) \times \dots \times U(1)$. The representation of this group is the same as the adjoint representation of this group on $\mathfrak{sl}_{3}/\mathfrak{b}$, where $\mathfrak{b}$ is upper triangular $3\times 3$ matrices with trace zero. |
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Apr 30 |
answered | are moduli stacks deligne-mumford stacks in general |
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Apr 29 |
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Zariski’s main theorem in the form of Grothendieck, universal properties Laumon and Moret-Bailly, Chapitre 16, particularly Corollaire 16.6.2. Although I agree with Angelo in spirit, it is certainly helpful to readers of an article to have a secondary source that they can refer to as necessary. |
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Apr 29 |
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Defining Equations of a Flag Variety Another keyword, in addition to "standard monomials", is "straightening law". |
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Apr 29 |
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Defining Equations of a Flag Variety I wholeheartedly agree with @Barbara. If you get stuck, you might check Griffiths and Harris, p. 211. They do the case of the Grassmannian, but the "diagonal" morphism of a flag variety into the product of Grassmannians is a closed immersion, and defining equations of that closed immersion are easier to compute. |
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Apr 26 |
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canonical model of a reducible curve @IMeasy: As Angelo points out, usually you need to take a sufficiently positive tensor power of the dualizing sheaf before the global sections define a closed immersion into projective space; I believe the third power always works. Mumford proved that for the closed immersion associated to the fifth power, the corresponding point of the Hilbert scheme is stable in the sense of Geometric Invariant Theory (you did not ask, but I thought I would mention it). |
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Apr 26 |
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canonical model of a reducible curve I am confused by this question. There is, indeed, a precise definition of the term "canonical model", and it is (usually) defined using the dualizing sheaf. Are you asking for a statement of that definition? Or are you asking for alternative definitions (perhaps more "natural" than the definition you already know)? Also, what do you mean by "twisted"? It is indeed true that the restriction of the dualizing sheaf of the total curve to a component is different from the dualizing sheaf of that component. Is that what you are referring to? |
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Apr 26 |
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Families of local rings coming from a locally ringed space Why do you ask this question? Is there a specific assignment of local rings that you are trying to "realize"? |
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Apr 23 |
revised |
line bundle on a nodal curve coming degeneration of smooth ones Corrected typo |
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Apr 23 |
answered | line bundle on a nodal curve coming degeneration of smooth ones |
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Apr 23 |
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line bundle on a nodal curve coming degeneration of smooth ones I believe you probably want to add the condition that the generic fiber of $\mathcal{X}$ is smooth. |
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Apr 18 |
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hyperelliptic stable genus four curve My suggestion is to do a case-by-case analysis. Using the theory of admissible covers, you can explicitly list all the dual graphs that can arise from a limit of a hyperelliptic curve. I went through a couple of these and proved that every "smoothable" $\mathfrak{g}^1_3$ is a limit of a "hyperelliptic" $\mathfrak{g}^1_3$. |
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Apr 18 |
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hyperelliptic stable genus four curve This argument is not correct. Although all smooth genus $4$ curves have "at least two" $\mathfrak{g}^1_3$s, or more precisely, a length $2$ subscheme of $\text{Pic}^3_X$ inside $W^1_3(X)$, nonetheless, every smooth hyperelliptic curve has infinitely many $\mathfrak{g}^1_3$s. More precisely, for a smooth, hyperelliptic curve $X$, $W^1_3(X)$ is a copy of $X$ (embedded by the Abel map, and then translated). All of these $\mathfrak{g}^1_3$s come from the hyperelliptic $\mathfrak{g}^1_2$ by adding a base point. |
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Apr 16 |
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A question about $R$-points of an complex reductive group. @Strasser: No, the argument above does not apply to the full center. My first answer (which was only up for a few minutes) was a counterexample for the full center $Z$ in $\textb{SL}_2$. |
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Apr 16 |
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lifts of maps to $\mathcal{M}_{1,1}$ Neither the Siegel upper half space nor the open subset of $\mathbb{P}(2,3)$ that you mention is the stack of elliptic curves. The stack of elliptic curves is, generically, a $\mathbb{Z}_2$-gerbe. |
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Apr 16 |
answered | A question about $R$-points of an complex reductive group. |
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Apr 12 |
answered | When is the determinant of the push-forward of an ample line bundle ample |
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Apr 12 |
answered | Question about relative Cartier divisor. |
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Apr 11 |
answered | Determinant of coherent sheaves |
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Apr 11 |
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Are period domains ever contractible @Jason S: I am also a Jason S! |
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Apr 11 |
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Weil restriction of abelian schemes along finite étale (resp. finite locally free) morphisms @Simon: Cf. Murre. |
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Apr 10 |
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Weil restriction of abelian schemes along finite étale (resp. finite locally free) morphisms @Ricky: Cf. Murre. |
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Apr 9 |
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de Rham complex of closed immersion between smooth schemes There is more than one mistake. Since you have imposed no hypotheses on $P$ and $Q$, why should there exist etale morphisms as specified? |
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Apr 9 |
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relative resolution of singularity Perhaps you should look at "semistable reduction" of Kempf, Knudsen, Mumford and Saint-Donat. I believe this is the best you can hope for without some additional hypotheses. |
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Apr 6 |
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Tricks to produce examples of hypersurfaces with index greater than $1$ @Pannekoek: My comment above (perhaps a bit cryptic) is an example of a birational, projective morphism between non-normal varieties for which Lang-Nishimura fails. |
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Apr 6 |
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Tricks to produce examples of hypersurfaces with index greater than $1$ @Pannekoek: Consider, for instance, a plane conic $X$ with affine equation $au^2 + bv^2 = 1$ for $a,b\in K$ such that $X$ has no degree $1$ zero-cycle (easy to determine using Legendre's theorem). Consider the morphism $f:X\to \mathbb{A}^3_K$ given by $f(u,v) = (au^2-1,v,uv)$. This maps the degree $2$ zero-cycle $Z(au^2-1,v)$ to a $K$-point $(0,0,0)$. On the open complements, this map is an isomorphism. |
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Apr 2 |
answered | Existence of smooth surfaces containing a curve |
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Apr 2 |
answered | Non Cohen-Macaulay varieties and Groebner degeneration |
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Apr 1 |
answered | normal bundle of hyperelliptic locus |
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Mar 29 |
answered | Flatness and intersections of Cohen-Macaulay subvarieties |
