zen-blog/public/cases/qwen2-vl/1_6.json

{
    "title": "Example: Document Parsing with Dense Formulas",
    "messages": [
                {
            "role": "User",
            "content": "Convert the content in the image to Markdown\n![image](http://qianwen-res.oss-accelerate-overseas.aliyuncs.com/Qwen2-VL/arxiv_sample.png)"
        },
        {
            "role": "Qwen2-VL",
            "content": "1. If $i > 0$, let $\\underline{d}'' := (i-1, d-i-l+1, l)$. Let $V\\_{\\underline{d}}$ and $V\\_{\\underline{d}''}$ be $r+1$-dimensional subspaces of $H^0(\\mathcal{L}\\_{\\underline{d}})$ and $H^0(\\mathcal{L}\\_{\\underline{d}''})$, respectively, such that $\\varphi\\_{\\underline{d}, \\underline{d}''}(V\\_{\\underline{d}}) = V\\_{\\underline{d}''}^{X\\_1, 0}$. Then the distributivity holds in $V\\_{\\underline{d}''}$ if and only if\n$$\n\\dim (V\\_{\\underline{d}}^{X\\_2, 0} + V\\_{\\underline{d}}^{X\\_3, 0}) - \\dim (V\\_{\\underline{d}''}^{X\\_2, 0} + V\\_{\\underline{d}''}^{X\\_3, 0}) = r + 1 - \\dim (V\\_{\\underline{d}''}^{X\\_1, 0} + V\\_{\\underline{d}''}^{X\\_2, 0} + V\\_{\\underline{d}''}^{X\\_3, 0})\n$$\n\n2. If $i + l \\leq d - 1$, let $\\underline{d}''' := (i, d-i-l-1, l+1)$. Let $V\\_{\\underline{d}}$ and $V\\_{\\underline{d}'''}$ be $r+1$-dimensional subspaces of $H^0(\\mathcal{L}\\_{\\underline{d}})$ and $H^0(\\mathcal{L}\\_{\\underline{d}'''})$, respectively, such that $\\varphi\\_{\\underline{d}''', \\underline{d}}(V\\_{\\underline{d}'''}) = V\\_{\\underline{d}}^{X\\_3, 0}$. Then the distributivity holds in $V\\_{\\underline{d}}$ if and only if\n$$\n\\dim (V\\_{\\underline{d}'''}^{X\\_1, 0} + V\\_{\\underline{d}'''}^{X\\_2, 0}) - \\dim (V\\_{\\underline{d}}^{X\\_1, 0} + V\\_{\\underline{d}}^{X\\_2, 0}) = r + 1 - \\dim (V\\_{\\underline{d}}^{X\\_1, 0} + V\\_{\\underline{d}}^{X\\_2, 0} + V\\_{\\underline{d}}^{X\\_3, 0}),\n$$\nwhere the maps $\\varphi\\_{\\underline{d}, \\underline{d}''}$ and $\\varphi\\_{\\underline{d}''', \\underline{d}}$ in Proposition 3.14 are the maps linking the corresponding sheaves. Another important result is Proposition 3.16, which establishes an inequality for any exact limit linear series. Specifically, our Proposition 3.16 says:\n\nLet $\\{(\\mathcal{L}\\_{\\underline{d}}, V\\_{\\underline{d}})\\}\\_{\\underline{d}}$ be an exact limit linear series of degree $d$ and dimension $r$. Then\n$$\n\\sum\\_{\\underline{d}} \\dim \\left( \\frac{V\\_{\\underline{d}}}{V\\_{\\underline{d}}^{X\\_1, 0} + V\\_{\\underline{d}}^{X\\_2, 0} + V\\_{\\underline{d}}^{X\\_3, 0}} \\right) \\geq r + 1.\n$$\n\nAs a consequence of Proposition 3.14 and Proposition 3.16, in Corollary 3.17, we get the following characterization of exact limit linear series satisfying the distributivity at each multidegree:\n\nLet $\\{(\\mathcal{L}\\_{\\underline{d}}, V\\_{\\underline{d}})\\}\\_{\\underline{d}}$ be an exact limit linear series of degree $d$ and dimension $r$. Then\n$$\n\\sum\\_{\\underline{d}} \\dim \\left( \\frac{V\\_{\\underline{d}}}{V\\_{\\underline{d}}^{X\\_1, 0} + V\\_{\\underline{d}}^{X\\_2, 0} + V\\_{\\underline{d}}^{X\\_3, 0}} \\right) = r + 1\n$$\nif and only if the distributivity holds in $V\\_{\\underline{d}}$ for any $\\underline{d}$.\n\nIn Section 3, we also study simple limit linear series. In Proposition 3.18, we characterize the multidegrees which determine all simple bases. In Proposition 3.20, following the construction in Muñoz [5], Proposition 4.4, we present a certain structure of linearly independent sections for any exact limit linear series satisfying the distributivity property. In propositions 3.21 and 3.22, we obtain some properties of that structure. These properties imply the existence of a simple basis (we give a proof of that in Proposition 3.23).\n\nWe conclude Section 3 with a characterization of simple limit linear series among the exact limit linear series. Specifically, our Corollary 3.24 says:\n\nLet $\\{(\\mathcal{L}\\_{\\underline{d}}, V\\_{\\underline{d}})\\}\\_{\\underline{d}}$ be an exact limit linear series of degree $d$ and dimension $r$. Then the following statements are equivalent:\n1. $\\{(\\mathcal{L}\\_{\\underline{d}}, V\\_{\\underline{d}})\\}\\_{\\underline{d}}$ is simple.\n2. $\\sum\\_{\\underline{d}} \\dim \\left( \\frac{V\\_{\\underline{d}}}{V\\_{\\underline{d}}^{X\\_1, 0} + V\\_{\\underline{d}}^{X\\_2, 0} + V\\_{\\underline{d}}^{X\\_3, 0}} \\right) = r + 1$.\n3. The distributivity holds in $V\\_{\\underline{d}}$ for any $\\underline{d}$.\n\nIn a subsequent work, given a refined limit linear series, we will construct all simple extensions by combining our Proposition 3.14 and the method of Muñoz [6] for the construction of all exact extensions."
        }
    ]
}