Return to Answer

2 edited notation to make statement more precise

1) $f$ is isomorphic in codimension $d$ if it is an isomorphism near any codimension $d$ point in either $X$ or $Y$. Equivalently, there exists closed subsets $Z\subseteq X$ and $W\subseteq Y$ such that ${\rm codim}_XZ\geq d+1$, ${\rm codim}_YW\geq d+1$, and $X\setminus Z\simeq f:X\setminus Z\overset{\simeq}{\longrightarrow} Y\setminus W$ is an isomorphism.

2) By the Theorem of the Base of Néron–Severi, if $f$ is proper of finite type, then `$N_1(X/Y)_{\mathbb Q}$` and $N^1(X/Y)_{\mathbb Q}$ are finite-dimensional vector spaces of the same dimension. This is actually more than you need, because even without the finite type assumption it is true that the intersection pairing `$N_1(X/Y)_{\mathbb Q}\times N^1(X/Y)_{\mathbb Q}\to {\mathbb Q}$` is non-degenerate.

1

1) $f$ is isomorphic in codimension $d$ if it is an isomorphism near any codimension $d$ point in either $X$ or $Y$. Equivalently, there exists closed subsets $Z\subseteq X$ and $W\subseteq Y$ such that ${\rm codim}_XZ\geq d+1$, ${\rm codim}_YW\geq d+1$, and $X\setminus Z\simeq Y\setminus W$.

2) By the Theorem of the Base of Néron–Severi, if $f$ is proper of finite type, then `$N_1(X/Y)_{\mathbb Q}$` and $N^1(X/Y)_{\mathbb Q}$ are finite-dimensional vector spaces of the same dimension. This is actually more than you need, because even without the finite type assumption it is true that the intersection pairing `$N_1(X/Y)_{\mathbb Q}\times N^1(X/Y)_{\mathbb Q}\to {\mathbb Q}$` is non-degenerate.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 24, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.924602210521698, "perplexity_flag": "head"}