Cohomology of tangent sheaf of a hypersurface












3












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Let $Xsubsetmathbb{P}^n$ be an irreducible and reduced hypersurface of degree $d$. How can one explicitly compute the dimension of the vector spaces $H^0(X,T_X),H^1(X,T_X),H^2(X,T_X)$? Here $T_X$ is the tangent sheaf of $X$.
For instance $h^0(X,T_X)$ gives the dimension of the automorphism group of $X$.






ag.algebraic-geometry sheaf-theory projective-geometry birational-geometry sheaf-cohomology







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    3












    $begingroup$


    Let $Xsubsetmathbb{P}^n$ be an irreducible and reduced hypersurface of degree $d$. How can one explicitly compute the dimension of the vector spaces $H^0(X,T_X),H^1(X,T_X),H^2(X,T_X)$? Here $T_X$ is the tangent sheaf of $X$.
    For instance $h^0(X,T_X)$ gives the dimension of the automorphism group of $X$.






    ag.algebraic-geometry sheaf-theory projective-geometry birational-geometry sheaf-cohomology







    share|cite|improve this question









    $endgroup$















      3












      3








      3





      $begingroup$


      Let $Xsubsetmathbb{P}^n$ be an irreducible and reduced hypersurface of degree $d$. How can one explicitly compute the dimension of the vector spaces $H^0(X,T_X),H^1(X,T_X),H^2(X,T_X)$? Here $T_X$ is the tangent sheaf of $X$.
      For instance $h^0(X,T_X)$ gives the dimension of the automorphism group of $X$.






      ag.algebraic-geometry sheaf-theory projective-geometry birational-geometry sheaf-cohomology







      share|cite|improve this question









      $endgroup$




      Let $Xsubsetmathbb{P}^n$ be an irreducible and reduced hypersurface of degree $d$. How can one explicitly compute the dimension of the vector spaces $H^0(X,T_X),H^1(X,T_X),H^2(X,T_X)$? Here $T_X$ is the tangent sheaf of $X$.
      For instance $h^0(X,T_X)$ gives the dimension of the automorphism group of $X$.






      ag.algebraic-geometry sheaf-theory projective-geometry birational-geometry sheaf-cohomology




      ag.algebraic-geometry sheaf-theory projective-geometry birational-geometry sheaf-cohomology






      share|cite|improve this question













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      asked Mar 5 at 17:43









      user125056user125056

      361




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          1 Answer
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          $begingroup$

          Use the normal sequence
          $$
          0 to T_X to T_{mathbb{P}^n}vert_X to N_{X/mathbb{P}^n} to 0,
          $$
          exact sequences
          $$
          0 to mathcal{O}_{mathbb{P}^n} to mathcal{O}_{mathbb{P}^n}(d) to i_*N_{X/mathbb{P}^n} to 0
          $$
          (we identify here $N_{X/mathbb{P}^n}$ with $mathcal{O}_X(d)$ and denote by $i$ the embedding $X to mathbb{P}^n$) and
          $$
          0 to T_{mathbb{P}^n}(-d) to T_{mathbb{P}^n} to i_*(T_{mathbb{P}^n}vert_X) to 0,
          $$
          and Borel-Bott-Weil Theorem to compute cohomology on $mathbb{P}^n$.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            I think that the first exact sequence you wrote holds only if $X$ is smooth. What if $X$ is singular?
            $endgroup$
            – user125056
            Mar 5 at 18:57






          • 2




            $begingroup$
            This is right (I missed the absence of the smoothness assumption in the question). In the singular case the first sequence is not exact in the right term, but its cokernel is not so hard to control. It is isomorphic to $mathcal{O}_Z(d)$, where $Z subset mathbb{P}^n$ is the subscheme defined by ${ partial F/partial x_0 = partial F/partial x_1 = dots partial F/partial x_n = 0 }$, where $F$ is the equation of $X$.
            $endgroup$
            – Sasha
            Mar 5 at 19:31










          • $begingroup$
            An addendum: this nice paper by Sernesi (arxiv.org/pdf/1306.3736.pdf) gives informations on how to control the deformations of a singular reduced hypersurface in terms of the local cohomology of the cokernel that Sasha was mentioning.
            $endgroup$
            – Enrico
            Mar 5 at 23:08











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          1 Answer
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          1 Answer
          1






          active

          oldest

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          active

          oldest

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          active

          oldest

          votes









          4












          $begingroup$

          Use the normal sequence
          $$
          0 to T_X to T_{mathbb{P}^n}vert_X to N_{X/mathbb{P}^n} to 0,
          $$
          exact sequences
          $$
          0 to mathcal{O}_{mathbb{P}^n} to mathcal{O}_{mathbb{P}^n}(d) to i_*N_{X/mathbb{P}^n} to 0
          $$
          (we identify here $N_{X/mathbb{P}^n}$ with $mathcal{O}_X(d)$ and denote by $i$ the embedding $X to mathbb{P}^n$) and
          $$
          0 to T_{mathbb{P}^n}(-d) to T_{mathbb{P}^n} to i_*(T_{mathbb{P}^n}vert_X) to 0,
          $$
          and Borel-Bott-Weil Theorem to compute cohomology on $mathbb{P}^n$.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            I think that the first exact sequence you wrote holds only if $X$ is smooth. What if $X$ is singular?
            $endgroup$
            – user125056
            Mar 5 at 18:57






          • 2




            $begingroup$
            This is right (I missed the absence of the smoothness assumption in the question). In the singular case the first sequence is not exact in the right term, but its cokernel is not so hard to control. It is isomorphic to $mathcal{O}_Z(d)$, where $Z subset mathbb{P}^n$ is the subscheme defined by ${ partial F/partial x_0 = partial F/partial x_1 = dots partial F/partial x_n = 0 }$, where $F$ is the equation of $X$.
            $endgroup$
            – Sasha
            Mar 5 at 19:31










          • $begingroup$
            An addendum: this nice paper by Sernesi (arxiv.org/pdf/1306.3736.pdf) gives informations on how to control the deformations of a singular reduced hypersurface in terms of the local cohomology of the cokernel that Sasha was mentioning.
            $endgroup$
            – Enrico
            Mar 5 at 23:08
















          4












          $begingroup$

          Use the normal sequence
          $$
          0 to T_X to T_{mathbb{P}^n}vert_X to N_{X/mathbb{P}^n} to 0,
          $$
          exact sequences
          $$
          0 to mathcal{O}_{mathbb{P}^n} to mathcal{O}_{mathbb{P}^n}(d) to i_*N_{X/mathbb{P}^n} to 0
          $$
          (we identify here $N_{X/mathbb{P}^n}$ with $mathcal{O}_X(d)$ and denote by $i$ the embedding $X to mathbb{P}^n$) and
          $$
          0 to T_{mathbb{P}^n}(-d) to T_{mathbb{P}^n} to i_*(T_{mathbb{P}^n}vert_X) to 0,
          $$
          and Borel-Bott-Weil Theorem to compute cohomology on $mathbb{P}^n$.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            I think that the first exact sequence you wrote holds only if $X$ is smooth. What if $X$ is singular?
            $endgroup$
            – user125056
            Mar 5 at 18:57






          • 2




            $begingroup$
            This is right (I missed the absence of the smoothness assumption in the question). In the singular case the first sequence is not exact in the right term, but its cokernel is not so hard to control. It is isomorphic to $mathcal{O}_Z(d)$, where $Z subset mathbb{P}^n$ is the subscheme defined by ${ partial F/partial x_0 = partial F/partial x_1 = dots partial F/partial x_n = 0 }$, where $F$ is the equation of $X$.
            $endgroup$
            – Sasha
            Mar 5 at 19:31










          • $begingroup$
            An addendum: this nice paper by Sernesi (arxiv.org/pdf/1306.3736.pdf) gives informations on how to control the deformations of a singular reduced hypersurface in terms of the local cohomology of the cokernel that Sasha was mentioning.
            $endgroup$
            – Enrico
            Mar 5 at 23:08














          4












          4








          4





          $begingroup$

          Use the normal sequence
          $$
          0 to T_X to T_{mathbb{P}^n}vert_X to N_{X/mathbb{P}^n} to 0,
          $$
          exact sequences
          $$
          0 to mathcal{O}_{mathbb{P}^n} to mathcal{O}_{mathbb{P}^n}(d) to i_*N_{X/mathbb{P}^n} to 0
          $$
          (we identify here $N_{X/mathbb{P}^n}$ with $mathcal{O}_X(d)$ and denote by $i$ the embedding $X to mathbb{P}^n$) and
          $$
          0 to T_{mathbb{P}^n}(-d) to T_{mathbb{P}^n} to i_*(T_{mathbb{P}^n}vert_X) to 0,
          $$
          and Borel-Bott-Weil Theorem to compute cohomology on $mathbb{P}^n$.






          share|cite|improve this answer









          $endgroup$



          Use the normal sequence
          $$
          0 to T_X to T_{mathbb{P}^n}vert_X to N_{X/mathbb{P}^n} to 0,
          $$
          exact sequences
          $$
          0 to mathcal{O}_{mathbb{P}^n} to mathcal{O}_{mathbb{P}^n}(d) to i_*N_{X/mathbb{P}^n} to 0
          $$
          (we identify here $N_{X/mathbb{P}^n}$ with $mathcal{O}_X(d)$ and denote by $i$ the embedding $X to mathbb{P}^n$) and
          $$
          0 to T_{mathbb{P}^n}(-d) to T_{mathbb{P}^n} to i_*(T_{mathbb{P}^n}vert_X) to 0,
          $$
          and Borel-Bott-Weil Theorem to compute cohomology on $mathbb{P}^n$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Mar 5 at 18:14









          SashaSasha

          20.9k22755




          20.9k22755








          • 1




            $begingroup$
            I think that the first exact sequence you wrote holds only if $X$ is smooth. What if $X$ is singular?
            $endgroup$
            – user125056
            Mar 5 at 18:57






          • 2




            $begingroup$
            This is right (I missed the absence of the smoothness assumption in the question). In the singular case the first sequence is not exact in the right term, but its cokernel is not so hard to control. It is isomorphic to $mathcal{O}_Z(d)$, where $Z subset mathbb{P}^n$ is the subscheme defined by ${ partial F/partial x_0 = partial F/partial x_1 = dots partial F/partial x_n = 0 }$, where $F$ is the equation of $X$.
            $endgroup$
            – Sasha
            Mar 5 at 19:31










          • $begingroup$
            An addendum: this nice paper by Sernesi (arxiv.org/pdf/1306.3736.pdf) gives informations on how to control the deformations of a singular reduced hypersurface in terms of the local cohomology of the cokernel that Sasha was mentioning.
            $endgroup$
            – Enrico
            Mar 5 at 23:08














          • 1




            $begingroup$
            I think that the first exact sequence you wrote holds only if $X$ is smooth. What if $X$ is singular?
            $endgroup$
            – user125056
            Mar 5 at 18:57






          • 2




            $begingroup$
            This is right (I missed the absence of the smoothness assumption in the question). In the singular case the first sequence is not exact in the right term, but its cokernel is not so hard to control. It is isomorphic to $mathcal{O}_Z(d)$, where $Z subset mathbb{P}^n$ is the subscheme defined by ${ partial F/partial x_0 = partial F/partial x_1 = dots partial F/partial x_n = 0 }$, where $F$ is the equation of $X$.
            $endgroup$
            – Sasha
            Mar 5 at 19:31










          • $begingroup$
            An addendum: this nice paper by Sernesi (arxiv.org/pdf/1306.3736.pdf) gives informations on how to control the deformations of a singular reduced hypersurface in terms of the local cohomology of the cokernel that Sasha was mentioning.
            $endgroup$
            – Enrico
            Mar 5 at 23:08








          1




          1




          $begingroup$
          I think that the first exact sequence you wrote holds only if $X$ is smooth. What if $X$ is singular?
          $endgroup$
          – user125056
          Mar 5 at 18:57




          $begingroup$
          I think that the first exact sequence you wrote holds only if $X$ is smooth. What if $X$ is singular?
          $endgroup$
          – user125056
          Mar 5 at 18:57




          2




          2




          $begingroup$
          This is right (I missed the absence of the smoothness assumption in the question). In the singular case the first sequence is not exact in the right term, but its cokernel is not so hard to control. It is isomorphic to $mathcal{O}_Z(d)$, where $Z subset mathbb{P}^n$ is the subscheme defined by ${ partial F/partial x_0 = partial F/partial x_1 = dots partial F/partial x_n = 0 }$, where $F$ is the equation of $X$.
          $endgroup$
          – Sasha
          Mar 5 at 19:31




          $begingroup$
          This is right (I missed the absence of the smoothness assumption in the question). In the singular case the first sequence is not exact in the right term, but its cokernel is not so hard to control. It is isomorphic to $mathcal{O}_Z(d)$, where $Z subset mathbb{P}^n$ is the subscheme defined by ${ partial F/partial x_0 = partial F/partial x_1 = dots partial F/partial x_n = 0 }$, where $F$ is the equation of $X$.
          $endgroup$
          – Sasha
          Mar 5 at 19:31












          $begingroup$
          An addendum: this nice paper by Sernesi (arxiv.org/pdf/1306.3736.pdf) gives informations on how to control the deformations of a singular reduced hypersurface in terms of the local cohomology of the cokernel that Sasha was mentioning.
          $endgroup$
          – Enrico
          Mar 5 at 23:08




          $begingroup$
          An addendum: this nice paper by Sernesi (arxiv.org/pdf/1306.3736.pdf) gives informations on how to control the deformations of a singular reduced hypersurface in terms of the local cohomology of the cokernel that Sasha was mentioning.
          $endgroup$
          – Enrico
          Mar 5 at 23:08


















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