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edited Nov 8 2010 at 20:48 |
Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = f^{-1}(\kappa)$ for small $\kappa > 0$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f,0}$, and let $p_{g} = \dim H^{n-2}(\tilde{V}, \mathcal{O})$ denote the corresponding geometric genus of $V$, which is independent of the chosen resolution. (This is the same geometric genus in the Yau-Durfee Theorem: $\mu \geq n! \ p_{g}$ for $n > 2$, See Y93 or Y06.)
Define the map $\varphi_{f} = f/\| f \| \colon S_{\epsilon}^{2n-1} \setminus V_{f, \kappa} \to S^{1}$, where $\epsilon > 0$ is sufficiently small. The intersection of $V_{\kappa}$ with a small sphere $S_{\epsilon}^{2n-1}$ is an algebraic link.
Consider the Briesksorn-Pham polynomials Brieskorn-Pham polynomial $f = \sum_{i = 1}^{n} z_{i}^{a_{i}}$ with $a_{i} \geq 2$. For In particular, for $f = z_{1}^{p} + z_{2}^{q}$, the intersection above is a torus knot $T_{p,q}$ if $p$ and $q$ are coprime. In 1968, Milnor conjectured that the unknotting number $u(T_{p,q})$ is related to the dimension of the local algebra $A_{f} = \mathbb{C}[x,y] / \langle x^{p-1}, y^{q-1} \rangle$, which is also the rank of the middle homology group $H_{1}(F_{f,0}; \mathbb{Z})$ of the corresponding fiber $F_{f,0} = \varphi_{f}^{-1}(e^{i \theta})$.
In 1928, Brauner proved $\mu(f) = \text{rank} \ H_{1}(F_{f,0}, \mathbb{Z}) = (p-1)(q-1)$ for $f = x^{p} + y^{q}$. In 1994, Kronheimer and Mrowka proved the Thom conjecture, and as a consequence, the Milnor conjecture followed.
(Edited) Question(s): In general, how is Is the genus $g$ of the fiber $F_{f,0}$ related to the geometric genus $p_{g}$ of $V$ for $n > 2$? References are welcome.Is there a geometric genus-like object for $n = 2$? If so, how might it relate to invariants of the torus knots?
Thanks!
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edited Nov 8 2010 at 20:35 |
Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = f^{-1}(\kappa)$ for small $\kappa > 0$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f,0}$, and let $p_{g} = \dim H^{n-2}(\tilde{V}, \mathcal{O})$ denote the corresponding geometric genus of $V$, which is independent of the chosen resolution. (This is the same geometric genus in the Yau-Durfee Theorem: $\mu \geq n! \ p_{g}$ for $n > 2$, See Y93 or Y06.)
Define the map $\varphi_{f} = f/\| f \| \colon S_{\epsilon}^{2n-1} \setminus V_{f, \kappa} \to S^{1}$, where $\epsilon > 0$ is sufficiently small. The intersection of $V_{\kappa}$ with a small sphere $S_{\epsilon}^{2n-1}$ is an algebraic link.
Consider the Briesksorn-Pham polynomials $f = \sum_{i = 1}^{n} z_{i}^{a_{i}}$ with $a_{i} \geq 2$. For $f = z_{1}^{p} + z_{2}^{q}$, the intersection above is a torus knot $T_{p,q}$ if $p$ and $q$ are coprime. In 1968, Milnor conjectured that the unknotting number $u(T_{p,q})$ is related to the dimension of the local algebra $A_{f} = \mathbb{C}[x,y] / \langle x^{p-1}, y^{q-1} \rangle$, which is also the rank of the middle homology group $H_{1}(F_{f,0}; \mathbb{Z})$ of the corresponding fiber $F_{f,0} = \varphi_{f}^{-1}(e^{i \theta})$.
In 1928, Brauner proved $\mu(f) = \text{rank} \ H_{1}(F_{f,0}, \mathbb{Z}) = (p-1)(q-1)$ for $f = x^{p} + y^{q}$. In 1994, Kronheimer and Mrowka proved the Thom conjecture, and as a consequence, the Milnor conjecture followed.
Question(s): In general, how is the genus $g$ of the fiber $F_{f,0}$ related to the geometric genus $p_{g}$ of $V$? Are there sufficient and necessary conditions on the geometric genus of $V$ which imply the Milnor conjecture for the case $f = z_{1}^{p} + z_{2}^{q}$n > 2$? References are welcome.
Thanks!
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edited Nov 8 2010 at 19:15 |
Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = f^{-1}(\kappa)$ for small $\kappa > 0$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f,0}$, and let $p_{g} = \dim H^{n-2}(\tilde{V}, \mathcal{O})$ denote the corresponding geometric genus of $V$, which is independent of the chosen resolution. (This is the same geomtric geometric genus in the Yau-Durfee Theorem: $\mu \geq n! \ p_{g}$ for $n > 2$, See Y93 or Y06.)
Define the map $\varphi_{f} = f/\| f \| \colon S_{\epsilon}^{2n-1} \setminus V_{f, \kappa} \to S^{1}$, where $\epsilon > 0$ is sufficiently small. The intersection of $V_{\kappa}$ with a small sphere $S_{\epsilon}^{2n-1}$ is an algebraic link.
Consider the Briesksorn-Pham polynomials $f = \sum_{i = 1}^{n} z_{i}^{a_{i}}$ with $a_{i} \geq 2$. For $f = z_{1}^{p} + z_{2}^{q}$, the intersection above is a torus knot $T_{p,q}$ if $p$ and $q$ are coprime. In 1968, Milnor conjectured that the unknotting number $u(T_{p,q})$ is related to the dimension of the local algebra $A_{f} = \mathbb{C}[x,y] / \langle x^{p-1}, y^{q-1} \rangle$, which is also the rank of the middle homology group $H_{1}(F_{f,0}; \mathbb{Z})$ of the corresponding fiber $F_{f,0} = \varphi_{f}^{-1}(e^{i \theta})$.
In 1928, Brauner proved $\mu(f) = \text{rank} \ H_{1}(F_{f,0}, \mathbb{Z}) = (p-1)(q-1)$ for $f = x^{p} + y^{q}$. In 1994, Kronheimer and Mrowka proved the Thom conjecture, and as a consequence, the Milnor conjecture followed.
Question(s): In general, how is the genus $g$ of the fiber $F_{f,0}$ related to the geometric genus $p_{g}$ of $V$? Are there sufficient and necessary conditions on the geometric genus of $V$ which imply the Milnor conjecture for the case $f = z_{1}^{p} + z_{2}^{q}$? References are welcome.
Thanks!
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edited Nov 8 2010 at 19:03 |
Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = f^{-1}(\kappa)$ for small $\kappa > 0$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f,0}$, and let $p_{g} = \dim H^{n-2}(\tilde{V}, \mathcal{O})$ denote the corresponding geometric genus of $V$, which is independent of the chosen resolution. (This is the same geomtric genus in the Yau-Durfee Theorem: $\mu \geq n! \ p_{g}$ for $n > 2$, See http://www.intlpress.com/JDG/archive/1993/37-2-375.pdf)Y93 or Y06.)
Define the map $\varphi_{f} = f/\| f \| \colon S_{\epsilon}^{2n-1} \setminus V_{f, \kappa} \to S^{1}$, where $\epsilon > 0$ is sufficiently small. The intersection of $V_{\kappa}$ with a small sphere $S_{\epsilon}^{2n-1}$ is an algebraic link.
Consider the Briesksorn-Pham polynomials $f = \sum_{i = 1}^{n} z_{i}^{a_{i}}$ with $a_{i} \geq 2$. For $f = z_{1}^{p} + z_{2}^{q}$, the intersection above is a torus knot $T_{p,q}$ if $p$ and $q$ are coprime. In 1968, Milnor conjectured that the unknotting number $u(T_{p,q})$ is related to the dimension of the local algebra $A_{f} = \mathbb{C}[x,y] / \langle x^{p-1}, y^{q-1} \rangle$, which is also the rank of the middle homology group $H_{1}(F_{f,0}; \mathbb{Z})$ of the corresponding fiber $F_{f,0} = \varphi_{f}^{-1}(e^{i \theta})$.
In 1928, Brauner proved $\mu(f) = \text{rank} \ H_{1}(F_{f,0}, \mathbb{Z}) = (p-1)(q-1)$ for $f = x^{p} + y^{q}$. In 1994, Kronheimer and Mrowka proved the Thom conjecture, and as a consequence, the Milnor conjecture followed.
Question(s): In general, how is the genus $g$ of the fiber $F_{f,0}$ related to the geometric genus $p_{g}$ of $V$? Are there sufficient and necessary conditions on the geometric genus of $V$ which imply the Milnor conjecture for the case $f = z_{1}^{p} + z_{2}^{q}$? References are welcome.
Thanks!
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edited Nov 8 2010 at 18:56 |
Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = f^{-1}(\kappa)$ for small $\kappa > 0$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f,0}$, and let $p_{g} = \dim H^{n-2}(\tilde{V}, \mathcal{O})$ denote the corresponding geometric genus of $V$, which is independent of the chosen resolution. (This is the same geomtric genus in the Yau-Durfee Theorem: $\mu \geq n! \ p_{g}$ for $n > 2$, See http://www.intlpress.com/JDG/archive/1993/37-2-375.pdf).
Define the map $\varphi_{f} = f/\| f \| \colon S_{\epsilon}^{2n-1} \setminus V_{f, \kappa} \to S^{1}$, where $\epsilon > 0$ is sufficiently small. The intersection of $V_{\kappa}$ with a small sphere $S_{\epsilon}^{2n-1}$ is an algebraic link.
Consider the Briesksorn-Pham polynomials $f = \sum_{i = 1}^{n} z_{i}^{a_{i}}$ with $a_{i} \geq 2$. For $f = z_{1}^{p} + z_{2}^{q}$, the intersection above is a torus knot $T_{p,q}$ if $p$ and $q$ are coprime. In 1968, Milnor conjectured that the unknotting number $u(T_{p,q})$ is related to the dimension of the local algebra $A_{f} = \mathbb{C}[x,y] / \langle x^{p-1}, y^{q-1} \rangle$, which is also the rank of the middle homology group $H_{1}(F_{f,0}; \mathbb{Z})$ of the corresponding fiber $F_{f,0} = \varphi_{f}^{-1}(e^{i \theta})$.
In 1928, Brauner proved $\mu(f) = \text{rank} \ H_{1}(F_{f,0}, \mathbb{Z}) = (p-1)(q-1)$ for $f = x^{p} + y^{q}$. In 1994, Kronheimer and Mrowka proved the Thom conjecture, and as a consequence, the Milnor conjecture followed.
Question(s): In general, how is the genus $g$ of the fiber $F_{f,0}$ related to the geometric genus $p_{g}$ of $V$? Are there sufficient and necessary conditions on the geometric genus of $V$ which imply the Milnor conjecture for the case $f = z_{1}^{p} + z_{2}^{q}$? References are welcome.
Thanks!
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edited Nov 8 2010 at 18:41 |
Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = f^{-1}(\kappa)$ for small $\kappa > 0$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f,0}$, and let $p_{g} = \dim H^{n-2}(\tilde{V}, \mathcal{O})$ denote the corresponding geometric genus of $V$, which is independent of the chosen resolution.
Define the map $\varphi_{f} = f/\| f \| \colon S_{\epsilon}^{2n-1} \setminus V_{f, \kappa} \to S^{1}$, where $\epsilon > 0$ is sufficiently small. The intersection of $V_{\kappa}$ with a small sphere $S_{\epsilon}^{2n-1}$ is an algebraic link.
Consider the Briesksorn-Pham polynomials $f = \sum_{i = 1}^{n} z_{i}^{a_{i}}$ with $a_{i} \geq 2$. For $f = z_{1}^{p} + z_{2}^{q}$, the intersection above is a torus knot $T_{p,q}$ if $p$ and $q$ are coprime. In 1968, Milnor conjectured that the unknotting number $u(T_{p,q})$ is related to the dimension of the local algebra $A_{f} = \mathbb{C}[x,y] / \langle x^{p-1}, y^{q-1} \rangle$, which is also the rank of the middle homology group $H_{1}(F_{f,0}; \mathbb{Z})$ of the corresponding fiber $F_{f,0} = \varphi_{f}^{-1}(e^{i \theta})$.
In 1928, Brauner proved $\mu(f) = \text{rank} \ H_{1}(F_{f,0}, \mathbb{Z}) = (p-1)(q-1)$ for $f = x^{p} + y^{q}$. In 1994, Kronheimer and Mrowka proved the Thom conjecture, and as a consequence, the Milnor conjecture followed.
Question(s): In general, how is the genus $g$ of the fiber $F_{f,0}$ related to the geometric genus $p_{g}$ of $V$? Are there sufficient and necessary conditions on the geometric genus of $V$ which imply the Milnor conjecture for the case $f = z_{1}^{p} + z_{2}^{q}$? References are welcome.
Thanks!
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edited Nov 8 2010 at 17:44 |
Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = f^{-1}(\kappa)$ for small $\kappa > 0$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f,0}$, and let $p_{g} = \dim H^{n-1}(\tilde{V}H^{n-2}(\tilde{V}, \mathcal{O})$ denote the corresponding geometric genus of $V$, which is independent of the chosen resolution.
Define the map $\varphi_{f} = f/\| f \| \colon S_{\epsilon}^{2n-1} \setminus V_{f, \kappa} \to S^{1}$, where $\epsilon > 0$ is sufficiently small. The intersection of $V_{\kappa}$ with a small sphere $S_{\epsilon}^{2n-1}$ is an algebraic link.
Consider the Briesksorn-Pham polynomials $f = \sum_{i = 1}^{n} z_{i}^{a_{i}}$ with $a_{i} \geq 2$. For $f = z_{1}^{p} + z_{2}^{q}$, the intersection above is a torus knot $T_{p,q}$ if $p$ and $q$ are coprime. In 1968, Milnor conjectured that the unknotting number $u(T_{p,q})$ is related to the dimension of local algebra $A_{f} = \mathbb{C}[x,y] / \langle x^{p-1}, y^{q-1} \rangle$, which is also the rank of the middle homology group $H_{1}(F_{f,0}; \mathbb{Z})$ of the corresponding fiber $F_{f,0} = \varphi_{f}^{-1}(e^{i \theta})$.
In 1928, Brauner proved $\mu(f) = \text{rank} \ H_{1}(F_{f,0}, \mathbb{Z}) = (p-1)(q-1)$ for $f = x^{p} + y^{q}$. In 1994, Kronheimer and Mrowka proved the Thom conjecture, and as a consequence, the Milnor conjecture followed.
Question(s): In general, how is the genus $g$ of the fiber $F_{f,0}$ related to the geometric genus $p_{g}$ of $V$? Are there sufficient and necessary conditions on the geometric genus of $V$ which imply the Milnor conjecture for the case $f = z_{1}^{p} + z_{2}^{q}$? References are welcome.
Thanks!
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edited Nov 8 2010 at 17:19 |
Let $f \colon (\mathbb{C}^{2},\mathbf{0}) \mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a Brieskorn-Pham singularity of the form $f = x^{p} + y^{q}$ complex analytic function with coprime $p,q > 1$isolated critical point at the origin. Define the singular hypersurface $V_{f} V_{f, \kappa} = f^{-1}(0)$f^{-1}(\kappa)$ for small $\kappa > 0$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f}$V_{f,0}$, and let $p_{g} = \dim H^{0}(VH^{n-1}(\tilde{V}, \mathcal{O})$ denote the corresponding geometric genus of $V$, which is independent of the chosen resolution.
Define the map $\varphi_{f} = f/\| f \| \colon S^{3}_{\epsilonS_{\epsilon}^{2n-1} \setminus V V_{f, \kappa} \to S^{1}$, where $\epsilon > 0$ is sufficiently small. The intersection of $V$ V_{\kappa}$ with a small sphere $S_{\epsilon}^{3}$ S_{\epsilon}^{2n-1}$ is an algebraic link.
Consider the Briesksorn-Pham polynomials $f = \sum_{i = 1}^{n} z_{i}^{a_{i}}$ with $a_{i} \geq 2$. For $f = z_{1}^{p} + z_{2}^{q}$, the intersection above is a torus link knot $T_{p,q}$. T_{p,q}$ if $p$ and $q$ are coprime. In 1968, Milnor conjectured that the unknotting number $u(T_{p,q})$ is related to the dimension of local algebra $A_{f} = \mathbb{C}[x,y] / \langle x^{p-1}, y^{q-1} \rangle$, which is also the rank of the middle homology group $H_{1}(F_{f,0}; \mathbb{Z})$ of the corresponding fiber $F_{f,0} = \varphi^{-1}(e^{i varphi_{f}^{-1}(e^{i \theta})$.
In 1928, Brauner proved $\mu(f) = \text{rank} \ H_{1}(F_{f,0}, \mathbb{Z}) = (p-1)(q-1)$ for $f = x^{p} + y^{q}$. In 1994, Kronheimer and Mrowka proved the Thom conjecture, and as a consequence, the Milnor conjecture followed.
Question(s): How In general, how is the genus $g$ of the fiber $f$ (viewed as a projective algebraic curve) F_{f,0}$ related to the geometric genus $p_{g}$ of $V$ (the hypersurface corresponding to the locus of zeros of $f$)? Do they determine each other? V$? Are there sufficient and necessary conditions on the geometric genus of $V$ which imply the Milnor conjecture for the case $f = z_{1}^{p} + z_{2}^{q}$? References are welcome.
Thanks!
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edited Nov 8 2010 at 16:50 |
Let $f \colon (\mathbb{C}^{2},\mathbf{0}) \to (\mathbb{C},0)$ be a Brieskorn-Pham singularity of the form $f = x^{p} + y^{q}$ with coprime $p,q > 1$. Define the singular hypersurface $V_{f} = f^{-1}(0)$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f}$, and let $p_{g} = \dim H^{0}(V, \mathcal{O})$ denote the corresponding geometric genus of $V$, which is independent of the chosen resolution. Define the map $\varphi_{f} = f/\| f \| \colon S^{3}_{\epsilon} \setminus V \to S^{1}$, where $\epsilon > 0$ is sufficiently small.
The intersection $V$ with a small sphere $S_{\epsilon}^{3}$ is a torus link $T_{p,q}$. In 1968, Milnor conjectured that the unknotting number $u(T_{p,q})$ is related to the dimension of local algebra $A_{f} = \mathbb{C}[x,y] / \langle x^{p-1}, y^{q-1} \rangle$, which is also the rank of the middle homology group $H_{1}(F_{f,0}; \mathbb{Z})$ of the corresponding fiber $F_{f,0} = \varphi^{-1}(e^{i \theta})$.
In 1928, Brauner proved $\mu(f) = \text{rank} \ H_{1}(F_{f,0}, \mathbb{Z}) = (p-1)(q-1)$ for $f = x^{p} + y^{q}$. In 1994, Kronheimer and Mrowka proved the Milnor Thom conjecture(, and as a consequenceof proving , the Thom Milnor conjecture with gauge theory) by first proving that the genus of projective algebraic curves of degree $d$ is bounded from below by that of the homogeneous curve of degree $d$ in the same homology classfollowed.
Question(s): How is the genus $g$ of $f$ (viewed as a projective algebraic curve) related to the geometric genus $p_{g}$ of $V$ (the hypersurface corresponding to the locus of zeros of $f$)? Do they determine each other? Is Are there a similar minimization of $p_{g}$ which implies the Milnor conjecture? What are sufficient and necessary conditions on the geometric genus which imply the Milnor conjecture? References are welcome.
Thanks!
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edited Nov 8 2010 at 16:15 |
Let $f \colon (\mathbb{C}^{2},\mathbf{0}) \to (\mathbb{C},0)$ be a Brieskorn-Pham singularity of the form $f = x^{p} + y^{q}$ with coprime $p,q > 1$. Define the singular hypersurface $V_{f} = f^{-1}(0)$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f}$, and let $p_{g} = \dim H^{0}(V, \mathcal{O})$ denote the corresponding geometric genus of $V$, which is independent of the chosen resolution. Define the map $\varphi_{f} = f/\| f \| \colon S^{3}_{\epsilon} \setminus V \to S^{1}$, where $\epsilon > 0$ is sufficiently small.
The intersection $V$ with a small sphere $S_{\epsilon}^{3}$ is a torus link $T_{p,q}$. In 1968, Milnor conjectured that the unknotting number $u(T_{p,q})$ is related to the dimension of local algebra $A_{f} = \mathbb{C}[x,y] / \langle x^{p-1}, y^{q-1} \rangle$, which is also the rank of the middle homology group $H_{1}(F_{f,0}; \mathbb{Z})$ of the corresponding fiber $F_{f,0} = \varphi^{-1}(e^{i \theta})$.
In 1928, Brauner proved $\mu(f) = \text{rank} \ H_{1}(F_{f,0}, \mathbb{Z}) = (p-1)(q-1)$. p-1)(q-1)$ for $f = x^{p} + y^{q}$. In 1994, Kronheimer and Mrowka proved the Milnor conjecture (as a consequence of proving the Thom conjecture with gauge theory) by first proving that the genus of projective algebraic curves of degree $d$ is bounded from below by that of the homogeneous curve of degree $d$ in the same homology class.
Question(s): How is the genus $g$ of $f$ (viewed as a projective algebraic curve) related to the geometric genus $p_{g}$ of $V$ (the hypersurface corresponding to the locus of zeros of $f$)? Do they determine each other? Is there a similar minimization of $p_{g}$ which implies the Milnor conjecture? What are sufficient and necessary conditions on the geometric genus which imply the Milnor conjecture? References are welcome.
Thanks!
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asked Nov 8 2010 at 16:08 |
Genera and the Milnor Conjecture on the Unknotting Number of a Torus Knot
Let $f \colon (\mathbb{C}^{2},\mathbf{0}) \to (\mathbb{C},0)$ be a Brieskorn-Pham singularity of the form $f = x^{p} + y^{q}$ with coprime $p,q > 1$. Define the singular hypersurface $V_{f} = f^{-1}(0)$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f}$, and let $p_{g} = \dim H^{0}(V, \mathcal{O})$ denote the corresponding geometric genus of $V$, which is independent of the chosen resolution. Define the map $\varphi_{f} = f/\| f \| \colon S^{3}_{\epsilon} \setminus V \to S^{1}$, where $\epsilon > 0$ is sufficiently small.
The intersection $V$ with a small sphere $S_{\epsilon}^{3}$ is a torus link $T_{p,q}$. In 1968, Milnor conjectured that the unknotting number $u(T_{p,q})$ is related to the dimension of local algebra $A_{f} = \mathbb{C}[x,y] / \langle x^{p-1}, y^{q-1} \rangle$, which is also the rank of the middle homology group $H_{1}(F_{f,0}; \mathbb{Z})$ of the corresponding fiber $F_{f,0} = \varphi^{-1}(e^{i \theta})$.
In 1928, Brauner proved $\mu(f) = \text{rank} \ H_{1}(F_{f,0}, \mathbb{Z}) = (p-1)(q-1)$. In 1994, Kronheimer and Mrowka proved the Milnor conjecture (as a consequence of proving the Thom conjecture with gauge theory) by first proving that the genus of projective algebraic curves of degree $d$ is bounded from below by that of the homogeneous curve of degree $d$ in the same homology class.
Question(s): How is the genus $g$ of $f$ (viewed as a projective algebraic curve) related to the geometric genus $p_{g}$ of $V$ (the hypersurface corresponding to the locus of zeros of $f$)? Do they determine each other? Is there a similar minimization of $p_{g}$ which implies the Milnor conjecture? What are sufficient and necessary conditions on the geometric genus which imply the Milnor conjecture? References are welcome.
Thanks!
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