Mixing
Classification of Complex Surfaces. Illustrations of the failure of genus to provide a good classification.
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Suppose that $X$ is a complex projective curve, i.e., a compact Riemann surface. In this case, a very useful invariant of such objects is the genus, a topological property of the manifold which signifies the number of holes in the Riemann surface.
In the minimal model program (which we may, for our purposes here restrict attention to surfaces) we look for a classification of projective varieties (over $mathbb{C}$) up to birational isomorphism. A useful invariant is the Kodaira dimension of the projective surface (see, e.g., https://en.wikipedia.org/wiki/Kodaira_dimension). The genus of a complex projective surface does not tell us much and does not provide a useful invariant for projective varieties of dimension greater than one.
My question is:
Are there any concrete illustrations which show how bad the genus is as a birational invariant of projective varieties $X$, where $dim_{mathbb{C}} X geq 2$?
algebraic-geometry complex-geometry projective-geometry
asked Jan 22 at 23:35
Kyle BroderKyle Broder
437
$endgroup$
add a comment |
$begingroup$
Suppose that $X$ is a complex projective curve, i.e., a compact Riemann surface. In this case, a very useful invariant of such objects is the genus, a topological property of the manifold which signifies the number of holes in the Riemann surface.
In the minimal model program (which we may, for our purposes here restrict attention to surfaces) we look for a classification of projective varieties (over $mathbb{C}$) up to birational isomorphism. A useful invariant is the Kodaira dimension of the projective surface (see, e.g., https://en.wikipedia.org/wiki/Kodaira_dimension). The genus of a complex projective surface does not tell us much and does not provide a useful invariant for projective varieties of dimension greater than one.
My question is:
Are there any concrete illustrations which show how bad the genus is as a birational invariant of projective varieties $X$, where $dim_{mathbb{C}} X geq 2$?
algebraic-geometry complex-geometry projective-geometry
asked Jan 22 at 23:35
Kyle BroderKyle Broder
437
$endgroup$
1
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That there are a bunch of non-isomorphic products of two elliptic curves having $g=1,P_g = 1$ ?
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– reuns
Jan 23 at 0:45
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@reuns What is your $P_g$ here? The Plurigenus?
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– Kyle Broder
Jan 23 at 1:29
1
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What definition of genus are you using for surfaces?
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– Mohan
Jan 23 at 2:28
add a comment |
$begingroup$
Suppose that $X$ is a complex projective curve, i.e., a compact Riemann surface. In this case, a very useful invariant of such objects is the genus, a topological property of the manifold which signifies the number of holes in the Riemann surface.
In the minimal model program (which we may, for our purposes here restrict attention to surfaces) we look for a classification of projective varieties (over $mathbb{C}$) up to birational isomorphism. A useful invariant is the Kodaira dimension of the projective surface (see, e.g., https://en.wikipedia.org/wiki/Kodaira_dimension). The genus of a complex projective surface does not tell us much and does not provide a useful invariant for projective varieties of dimension greater than one.
My question is:
Are there any concrete illustrations which show how bad the genus is as a birational invariant of projective varieties $X$, where $dim_{mathbb{C}} X geq 2$?
algebraic-geometry complex-geometry projective-geometry
asked Jan 22 at 23:35
Kyle BroderKyle Broder
437
$endgroup$
Suppose that $X$ is a complex projective curve, i.e., a compact Riemann surface. In this case, a very useful invariant of such objects is the genus, a topological property of the manifold which signifies the number of holes in the Riemann surface.
In the minimal model program (which we may, for our purposes here restrict attention to surfaces) we look for a classification of projective varieties (over $mathbb{C}$) up to birational isomorphism. A useful invariant is the Kodaira dimension of the projective surface (see, e.g., https://en.wikipedia.org/wiki/Kodaira_dimension). The genus of a complex projective surface does not tell us much and does not provide a useful invariant for projective varieties of dimension greater than one.
My question is:
Are there any concrete illustrations which show how bad the genus is as a birational invariant of projective varieties $X$, where $dim_{mathbb{C}} X geq 2$?
algebraic-geometry complex-geometry projective-geometry
algebraic-geometry complex-geometry projective-geometry
asked Jan 22 at 23:35
Kyle BroderKyle Broder
437
asked Jan 22 at 23:35
Kyle BroderKyle Broder
437
asked Jan 22 at 23:35
Kyle BroderKyle Broder
437
asked Jan 22 at 23:35
Kyle BroderKyle Broder
437
asked Jan 22 at 23:35
Kyle BroderKyle Broder
437
437
1
$begingroup$
That there are a bunch of non-isomorphic products of two elliptic curves having $g=1,P_g = 1$ ?
$endgroup$
– reuns
Jan 23 at 0:45
$begingroup$
@reuns What is your $P_g$ here? The Plurigenus?
$endgroup$
– Kyle Broder
Jan 23 at 1:29
1
$begingroup$
What definition of genus are you using for surfaces?
$endgroup$
– Mohan
Jan 23 at 2:28
add a comment |
1
$begingroup$
That there are a bunch of non-isomorphic products of two elliptic curves having $g=1,P_g = 1$ ?
$endgroup$
– reuns
Jan 23 at 0:45
$begingroup$
@reuns What is your $P_g$ here? The Plurigenus?
$endgroup$
– Kyle Broder
Jan 23 at 1:29
1
$begingroup$
What definition of genus are you using for surfaces?
$endgroup$
– Mohan
Jan 23 at 2:28
1
1
$begingroup$
That there are a bunch of non-isomorphic products of two elliptic curves having $g=1,P_g = 1$ ?
$endgroup$
– reuns
Jan 23 at 0:45
$begingroup$
That there are a bunch of non-isomorphic products of two elliptic curves having $g=1,P_g = 1$ ?
$endgroup$
– reuns
Jan 23 at 0:45
$begingroup$
@reuns What is your $P_g$ here? The Plurigenus?
$endgroup$
– Kyle Broder
Jan 23 at 1:29
$begingroup$
@reuns What is your $P_g$ here? The Plurigenus?
$endgroup$
– Kyle Broder
Jan 23 at 1:29
1
1
$begingroup$
What definition of genus are you using for surfaces?
$endgroup$
– Mohan
Jan 23 at 2:28
$begingroup$
What definition of genus are you using for surfaces?
$endgroup$
– Mohan
Jan 23 at 2:28
add a comment |
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$begingroup$
That there are a bunch of non-isomorphic products of two elliptic curves having $g=1,P_g = 1$ ?
$endgroup$
– reuns
Jan 23 at 0:45
$begingroup$
@reuns What is your $P_g$ here? The Plurigenus?
$endgroup$
– Kyle Broder
Jan 23 at 1:29
1
$begingroup$
What definition of genus are you using for surfaces?
$endgroup$
– Mohan
Jan 23 at 2:28